How To Label A Graph 303016862X, 9783030168629

Citation preview

SPRINGER BRIEFS IN MATHEMATICS

Gary Chartrand Cooroo Egan Ping Zhang

How to Label a Graph

123

SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa City, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York City, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York City, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA Editorial Board Member Luis Gustavo Nonato, São Carlos, Rio Grande do Sul, Brazil Paulo J. S. Silva, Campinas, São Paulo, Brazil

SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians.

BCAM SpringerBriefs Editorial Board Enrique Zuazua Deusto Tech Universidad de Deusto Bilbao, Spain and Departamento de Matemáticas Universidad Autónoma de Madrid Cantoblanco, Madrid, Spain Irene Fonseca Center for Nonlinear Analysis Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, USA Juan J. Manfredi Department of Mathematics University of Pittsburgh Pittsburgh, USA Emmanuel Trélat Laboratoire Jacques-Louis Lions Institut Universitaire de France Université Pierre et Marie Curie CNRS, UMR, Paris Xu Zhang School of Mathematics Sichuan University Chengdu, China BCAM SpringerBriefs aims to publish contributions in the following disciplines: Applied Mathematics, Finance, Statistics and Computer Science. BCAM has appointed an Editorial Board, who evaluate and review proposals. Typical topics include: a timely report of state-of-the-art analytical techniques, bridge between new research results published in journal articles and a contextual literature review, a snapshot of a hot or emerging topic, a presentation of core concepts that students must understand in order to make independent contributions. Please submit your proposal to the Editorial Board or to Francesca Bonadei, Executive Editor Mathematics, Statistics, and Engineering: [email protected].

More information about this series at http://www.springer.com/series/10030

Gary Chartrand Cooroo Egan Ping Zhang •

•

How to Label a Graph

123

Gary Chartrand Department of Mathematics Western Michigan University Kalamazoo, MI, USA

Cooroo Egan Melbourne, VIC, Australia

Ping Zhang Department of Mathematics Western Michigan University Kalamazoo, MI, USA

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-030-16862-9 ISBN 978-3-030-16863-6 (eBook) https://doi.org/10.1007/978-3-030-16863-6 Mathematics Subject Classification (2010): 05C05, 05C10, 05C15, 05C45, 05C70, 05C78, 05C90 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are solely and exclusively licensed, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Graph labelings have existed for well over a century. In fact, a graph G is often defined as a finite nonempty set V of objects called vertices and a set E of 2-element subsets of V called edges. This is further emphasized by expressing G as GðV; EÞ. If G has order n and size m, then the vertex set is often described as V ¼ VðGÞ ¼ fv1 ; v2 ; . . .; vn g and the edge set as E ¼ EðGÞ ¼ fe1 ; e2 ; . . .; em g. So, in a sense, the vertices of G are labeled as v1 ; v2 ; . . .; vn and the edges of G are labeled as e1 ; e2 ; . . .; em . The general topic of graph labelings is discussed in Chapter 1. Graph labelings as an area of research in graph theory didn’t really begin until the 1960s. Since then there have been numerous types of graph labelings introduced and studied. These have been described in a dynamic survey written by Joseph Gallian [18]. Indeed, prior to the current book, there have been five books written on graph labelings: 1. W. D. Wallis, Magic Graphs. Birkhäuser, Boston (2001). 2. M. Bača and M. Miller, Super Edge-Antimagic Graphs. BrownWalk Press, Boca Raton (2008). 3. A. M. Marr and W. D. Wallis, Magic Graphs, Second edition. Birkhäuser/ Springer, New York (2013). 4. M. Haviar and M. Ivaška, Vertex Labelings of Simple Graphs. Research and Exposition on Mathematics, Volume 34, Heldermann Verlag (2015). 5. S. C. López and F. A. Muntaner-Batle,Graceful, Harmonious and Magic Type Labelings: Relations and Techniques. SpringerBriefs in Mathematics. Springer, Cham (2017). Three of these books have therefore dealt with the topic of magic labelings and the related antimagic labelings. The concept of magic labelings was introduced by Jiří Sedláček in the Theory of Graphs and Its Applications—a Symposium held in Smolenice, Czechoslovakia in June 1963. The proceedings of this Symposium concluded with a list of 31 problems, #27 of which included the following definition by Sedláček [41]:

v

vi

Preface A connected graph G is magic if there exists a real-valued function on the edge set of G with the properties that (i) distinct edges have distinct labels and (ii) the sum of labels of the edges incident with each vertex of G is the same constant.

Three years later, the American mathematician Bonnie M. Stewart [47] wrote a paper on magic graphs. This paper was followed by other papers by Stewart [48] and Sedláček [42] on magic labelings. Seven years after the Symposium in Smolenice, Anton Kotzig (who also attended the Symposium) and Alexander Rosa [29] proposed another type of magic labeling, which is now commonly called an edge-magic total labeling. For a graph G of order n and size m, an edge-magic total labeling of G is a bijective function f : VðGÞ [ EðGÞ ! ½n þ m ¼ f1; 2; . . .; n þ mg such that f ðuvÞ þ f ðuÞ þ f ðvÞ is the same constant for every edge uv of G. If f ðvÞ plus the sum of the labels of the edges incident with v is the same constant for every vertex v of G, then f is a vertex-magic total labeling of G. The topic of magic labelings has been well covered, not only in the books mentioned above, but in numerous papers. Consequently, we will not discuss this area in the present book. Of the many graph labeling concepts and problems that have been introduced over the years, we have found some of particular interest to us that don’t appear to have received such wide recognition. This will then be the emphasis of this book. In 1966, Alexander Rosa [40] introduced a different view of vertex labeling, which later became known as a graceful labeling, when the vertices of a graph were labeled with distinct integers from a prescribed set that resulted in an edge labeling possessing a particular required property. This labeling is the primary topic in Chapter 2. In 1980, Ronald Graham and Neil Sloane [20] introduced another vertex labeling called a harmonious labeling where the vertex labels are selected from the ring Zm of integers modulo m, where m is the size of a graph, with the goal of producing an edge labeling in which edge labels are distinct. This is the primary topic in Chapter 3. Because the labels chosen for vertices in a graph labeling have often been positive integers, it is not surprising that some conditions placed on certain vertex labels have a number-theoretic flavor—in particular, requirements on the greatest common divisor of the labels of specified pairs of vertices in the graphs. Consequently, the prime number decompositions of labels in these labelings play an important role. Furthermore, some of the labelings can be expressed in terms of subsets of certain sets of integers. This is the topic in Chapter 4. As has already been noted, there have been several vertex labelings and edge labelings of graphs where each label is a positive integer and certain labels are added to produce a sum possessing a particular property. There are occasions when the resulting sums represent a coloring with a certain property. In these cases, the goal is often to minimize the largest label being used. This is the topic in Chapter 5.

Preface

vii

Chapter 6 concerns vertex labelings of planar graphs with labels taken from the ring Z3 of integers modulo 3. In this case, the labels of the vertices on the boundary of each zone (or region) are added to produce a label for the zone with the goal of arriving at a particular sum in Z3 . It is seen that there is a connection with these labelings and the famous Four Color Theorem. Kalamazoo, MI, USA Melbourne, Australia Kalamazoo, MI, USA February 2019

Gary Chartrand Cooroo Egan Ping Zhang

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Labeled Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Graph Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3

. . . .

7 7 12 16

3 Harmonious Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Harmonious Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Harmonious Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 25

4 Prime Labelings . . . . . . . . 4.1 Prime Graphs . . . . . . . 4.2 Multi-prime Labelings 4.3 Subset Labelings . . . . 4.4 2-Prime Labelings . . .

2 Graceful Labelings . . . . . . 2.1 Graceful Graphs . . . . . 2.2 Cyclic Decompositions 2.3 Graceful Trees . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

29 29 35 38 45

5 Additive Labelings . . . . . . . . . . . . . . . 5.1 Rainbow Additive Labelings . . . . . 5.2 Proper Additive Labelings . . . . . . . 5.3 Monochromatic Additive Labelings 5.4 Proper Sigma Labelings . . . . . . . . 5.5 Rainbow Sigma Labelings . . . . . . . 5.6 Monochromatic Sigma Labelings . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

53 53 59 60 63 65 66

6 Zonal Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Zonal Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Zonal Labelings of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 74

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

ix

x

Contents

6.3 Zonal Labelings of Cubic Maps . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Inner Zonal Labelings of Bicubic Maps . . . . . . . . . . . . . . . . . . . .

78 82

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Fig. 2.10 Fig. Fig. Fig. Fig. Fig. Fig.

2.11 2.12 2.13 3.1 3.2 3.3

Fig. Fig. Fig. Fig. Fig. Fig.

3.4 3.5 3.6 3.7 3.8 3.9

The trees of order 4 or 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . The labeled trees of order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . The labeled graphs of order 3 . . . . . . . . . . . . . . . . . . . . . . . . . A vertex-labeled graph and an edge-labeled graph . . . . . . . . . A vertex-labeled graph G1 and an edge-labeled graph G2 . . . A graceful graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Showing that a graph G is graceful . . . . . . . . . . . . . . . . . . . . The Petersen graph is graceful . . . . . . . . . . . . . . . . . . . . . . . . Showing that the graph G of Figure 2.1 is graceful . . . . . . . . The graphs C8 , C82 , and C83 . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclically decomposing K7 into seven copies of K3 . . . . . . . . Cyclically decomposing C93 into nine copies of K3 . . . . . . . . . A cyclic decomposition of K11 into copies of C5 . . . . . . . . . . 5 5 and C15 into Illustrating cyclic decompositions of C13 copies of C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A cyclical decomposition of K13 into copies of the bow tie graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The six trees of order 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The graph K6 and the graceful trees T1 ; T2 ; . . .; T6 . . . . . . . . . Graceful labelings of P7 , K1;6 , and S3;4 . . . . . . . . . . . . . . . . . . Harmonious labelings of K3 and K4 . . . . . . . . . . . . . . . . . . . . The Petersen graph is harmonious . . . . . . . . . . . . . . . . . . . . . The complete graph Km labeled with elements of Zm for m ¼ 6 and m ¼ 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three harmonious graphs of size 6 having different orders . . . Harmonious labelings of C5 and C7 . . . . . . . . . . . . . . . . . . . . Harmonious labelings of Pn for n ¼ 6; 7; 8; 9 and K1;6 . . . . . . Trees T for which hðTÞ ¼ jEðTÞj . . . . . . . . . . . . . . . . . . . . . . Harmonious graphs of sizes 2; 3; and 4 . . . . . . . . . . . . . . . . . Harmonious graphs of size 5 that are not trees . . . . . . . . . . . .

. . . . . . . . . . . . .

2 2 3 3 4 9 10 10 11 11 13 13 14

..

15

. . . . . .

. . . . . .

15 16 17 18 22 23

. . . . . . .

. . . . . . .

23 24 24 25 26 27 28

. . . . . . . . . . . . .

xi

xii

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

List of Figures

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22

Fig. 5.1 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 6.1 6.2 6.3 6.4 6.5 6.6

A prime graph Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A prime labeling of K3;6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . All trees of order 6 are prime . . . . . . . . . . . . . . . . . . . . . . . . . Graphs of order n and size 2dn=2e þ 1 that are not prime . . . gcd-rainbow labelings of K3 ; C5 , and P6 . . . . . . . . . . . . . . . . . Multi-prime labelings of C3 and C4 . . . . . . . . . . . . . . . . . . . . A multi-prime labeling of C5 . . . . . . . . . . . . . . . . . . . . . . . . . A graph G with qðGÞ ¼ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . The Petersen graph P has qðPÞ ¼ 5 . . . . . . . . . . . . . . . . . . . . Subset labelings of P4 , P5 , and P6 . . . . . . . . . . . . . . . . . . . . . Subset labelings of Pn for n ¼ 7; 8; 9 . . . . . . . . . . . . . . . . . . . A subset labeling of P11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subset labelings of C6 and C7 . . . . . . . . . . . . . . . . . . . . . . . . Two graphs with multi-prime index 3. . . . . . . . . . . . . . . . . . . Labeling the vertices of C5 and W5 ¼ C4 _ K1 . . . . . . . . . . . . A graph with no 2-prime labeling . . . . . . . . . . . . . . . . . . . . . . The nine forbidden induced subgraphs of a line graph . . . . . . A graph and its line graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-subset labelings of C5 and W5 ¼ C4 _ K1 . . . . . . . . . . . . . . A 3-prime labeling of the Petersen graph . . . . . . . . . . . . . . . . 3-subset labelings of the nine graphs in Figure 4.17 . . . . . . . . Two graphs G with wðGÞ ¼ 4 but wðG  vÞ ¼ 3 for every non-cut-vertex v of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connected graphs having exactly two vertices of the same degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irregular multigraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The graphs G1 ; G2 ; G3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rainbow additive labelings of G4 and G5 . . . . . . . . . . . . . . . . Rainbow additive labelings of K4 and K5 . . . . . . . . . . . . . . . . A rainbow additive labeling of K3;3 . . . . . . . . . . . . . . . . . . . . A rainbow additive labeling of K3;4 . . . . . . . . . . . . . . . . . . . . A tree T of order 19 with raðTÞ ¼ 19 3þ 2 ¼ 7. . . . . . . . . . . . . . Graphs with proper additive indices 1, 2, and 3 . . . . . . . . . . . A regular graph with a monochromatic additive labeling . . . . Monochromatic additive labelings of K2;3;4 and K3;4;5 . . . . . . Sigma labelings of C5 and C6 . . . . . . . . . . . . . . . . . . . . . . . . . Rainbow sigma labelings of C5 and C6 . . . . . . . . . . . . . . . . . A monochromatic sigma labeling of K2;3;4 . . . . . . . . . . . . . . . The six trees of order 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The graph K3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some n-cycles Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The planar graphs K4  e and K4 . . . . . . . . . . . . . . . . . . . . . . The graph C3 h K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A cubic plane graph that has no zonal labeling . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

30 31 32 33 34 35 36 38 39 40 41 41 43 45 46 46 47 48 48 49 50

..

51

. . . . . . . . . . . . . . . . . . . .

54 54 54 55 56 57 58 59 60 61 62 64 65 66 70 70 71 71 71 72

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

List of Figures

Fig. 6.7 Fig. Fig. Fig. Fig. Fig.

6.8 6.9 6.10 6.11 6.12

Fig. 6.13

Cubic bridgeless plane graphs and multigraphs of order at most 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proceeding about a cycle in a counterclockwise direction . . . . The type of a vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zonal labelings of a 3-cycle and a 4-cycle . . . . . . . . . . . . . . . The 3-colorings of a 3-cycle and a 4-cycle . . . . . . . . . . . . . . . The vertex types in a proper 3-coloring of the edges of a cubic map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A proper 3-coloring of the edges and a zonal labeling of Q3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

. . . . .

73 75 75 76 76

..

78

..

79

. . . . .

Chapter 1

Introduction

Graph labelings have been traced back to the 19th century when the famous British mathematician Arthur Cayley proved that there are n n−2 distinct labeled trees of order n (Cayley’s Tree Formula). During the past several decades, this topic has become a popular area of research in graph theory. In this chapter, an introduction to labeled graphs and graph labelings is presented. A discussion of different interpretations of labeling graphs is given here as well.

1.1 Labeled Graphs Arthur Cayley was a famous 19th century British mathematician. Among his many accomplishments, he was the first to define “group” in a modern way, namely as a set with a binary operation that satisfies certain laws. In fact, Cayley’s theorem in algebra states that every group is isomorphic to a group of permutations. While Cayley is undoubtedly best known for his work in modern algebra, he also worked in graph theory and proved a theorem in graph theory that also bears his name. A tree is a connected graph without cycles. Figure 1.1 shows all five trees of order 4 or 5, that is, having four or five vertices. None of these trees are labeled, that is, no vertex is assigned a label. Consequently, every tree of order 4 or 5 is isomorphic to one of the trees in Figure 1.1. Suppose that two trees T1 and T2 of order n are labeled with the same set of n labels, say 1, 2, . . . , n. Then T1 and T2 are considered the same if they have the same set of edges (necessarily n − 1 edges). The distinct labeled trees of order 4 are shown in Figure 1.2. Therefore, while there are 16 distinct labeled trees of order 4, there are only two non-isomorphic trees of order 4. We saw in Figure 1.1 that there are only three non-isomorphic trees of order 5. However, there are 125 distinct labeled trees of order 5. Cayley proved the following in [5]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6_1

1

2

1 Introduction

Fig. 1.1 The trees of order 4 or 5

Fig. 1.2 The labeled trees of order 4

Theorem 1.1 (Cayley’s Tree Formula) For each positive integer n, there are n n−2 distinct labeled trees of order n. One of the best known proofs of this formula is due to Heinz Prüfer [36]. Indeed, there are many different proofs of Cayley’s Tree Formula. John Moon [33] gave ten different proofs of this formula. For a labeled graph G of order n, whose n vertices are labeled 1, 2, . . . , n, there  are two possibilities for each of the n2 pairs i, j of distinct vertices of G, namely either G contains the edge i j or it doesn’t. This results in the following. Observation 1.2 For each positive integer n, there are 2(2) distinct labeled graphs of order n. n

3 Consequently, there are 2(2) = 23 = 8 distinct labeled graphs of order 3. All of these graphs are shown in Figure 1.3. By Cayley’s Tree Formula, three of these eight graphs are trees.

1.2 Graph Labelings

3

Fig. 1.3 The labeled graphs of order 3

1.2 Graph Labelings Over the years, there have been different interpretations as to what is meant by labeling a graph. We employ the most general definition here. A graph labeling is an assignment of labels (elements of some set) to elements of a graph G, usually the vertices or the edges (or both) of G. If it is only the vertices of G that are labeled, then the resulting graph is a vertex-labeled graph. In the case of edges, the resulting graph is edge-labeled. Traditionally, a vertex labeling of a graph required distinct vertices to be assigned distinct labels. Typically, these labels were positive integers. So, if G has order n, the vertex labels were often chosen to be the elements of the set [n] = {1, 2, . . . , n}. Similarly, an edge labeling of a graph G of size m ≥ 1 assigns distinct labels from the set [m] = {1, 2, . . . , m} to the edges of G. More recently, however, there has been no requirement that the labels in a graph labeling be positive integers or be distinct. Figure 1.4 shows a vertex-labeled graph and an edge-labeled graph, where the labels in the first graph are nonnegative integers and the labels in the second graph are elements of Z4 (the integers modulo 4). In the first graph, two vertices are labeled the same; in the second graph, two pairs of edges are labeled the same. In certain situations, it may be convenient for the labels to be colors, in which case, the labeling is a coloring. Traditionally, by a coloring of a graph G was meant an assignment of colors (elements of some set) to the vertices (or edges) of a graph G so that adjacent vertices (or edges) are assigned distinct colors. These are commonly called proper colorings. The major parameters dealing with proper colorings are the chromatic number χ (G) of a graph G, defined as the minimum number of colors in

Fig. 1.4 A vertex-labeled graph and an edge-labeled graph

4

1 Introduction

Fig. 1.5 A vertex-labeled graph G 1 and an edge-labeled graph G 2

a proper coloring of the vertices of G, and the chromatic index χ  (G), which is the minimum number of colors in a proper coloring of the edges of G. Over the years, this requirement has changed as well. For example, there has often been research dealing with rainbow colorings (both vertices and edges) where every two vertices (or edges) are required to be colored differently. That is, a rainbow coloring of the vertices of a graph G is equivalent to a traditional labeling of the vertices of G. Figure 1.5 shows a vertex-labeled graph G 1 where each vertex is labeled with its degree and an edge-labeled graph G 2 in which each edge of G 2 is assigned a color (r for red, b for blue, and g for green). Thus, G 2 is an edge-colored graph, where this coloring is being illustrated by means of an edge-labeled graph. Since no two adjacent edges of G 2 are colored the same, the coloring is a proper coloring. It is also possible to have a graph in which all vertices (or edges) are colored the same. These are called monochromatic colorings. It is quite possible then to have a graph labeling in which vertices (or edges) are labeled the same and quite possible to have a coloring in which adjacent vertices (or adjacent edges) are colored the same. In any case, it must be made clear what conditions are being placed on any labeling or coloring being considered. In either case though, the reason for studying any labeling or coloring is to satisfy some prescribed conditions. In the area of Ramsey Theory within graph theory, it is often the case that the edges of certain graphs (typically complete graphs) are arbitrarily colored with one of two colors (usually red or blue, resulting in a red-blue coloring) without any restriction whatsoever on the coloring, with the goal of looking for certain subgraphs all of whose edges are colored with a single color (that is, every edge of the subgraph is labeled the same), resulting in what is called a monochromatic subgraph. More precisely, when we refer to a vertex labeling of a graph G, it is meant that there exists a function f : V (G) → S for some set S. The function f itself is called the vertex labeling of G and the elements f (v) for v ∈ V (G) are vertex labels. These labels may or may not be distinct and the set S may or may not be a set of positive integers. As we will see, the set S may be a set of nonnegative integers, S may be the set Zk of integers modulo k for some integer k ≥ 2, or S may be some other set. While it is customary for S to be a set whose elements can be added or subtracted, no such restriction on the elements of S is required in general. Depending on the purpose for labeling the vertices of a graph, the labels themselves may be given other names. As mentioned above, if the labels are called colors, then

1.2 Graph Labelings

5

the labeling is a coloring; while if the labels are called weights, then the labeling is a weighting. Furthermore, if the labels are values of some sort, then the resulting labeling is a valuation; while if the labels are numbers of some kind, then the labeling is a numbering. There are also occasions when labels may be referred to as costs. All of this applies as well to labeling edges. In any graph theory problem involving a vertex labeling, there is always a purpose for labeling the vertices. Often the goal of a vertex labeling f of a graph G is its use in creating an edge labeling f  that has some prescribed property. Indeed, this process can be reversed by beginning with an edge labeling f of G and using it to generate a vertex labeling f  having a property of interest. Situations even more general than this are possible as we will see. Over the years, a number of graph labelings have been introduced, some of which have led to curious and unexpected problems that have defied numerous attempts to solve them. In the chapters that follow, we will describe several of these labelings and resulting problems.

Chapter 2

Graceful Labelings

The vertex labeling that very well may have had the greatest influence on the development of graph labelings as a research area in graph theory is the graceful labeling. Some historical background of this concept is described in this chapter as well as its applications in communications networks. Much research on graceful labeling has been focused on determining which graphs have a graceful labeling and its connection with graph decompositions. Major results and conjectures on graceful labelings are presented, including the most famous conjecture on this topic dealing with trees.

2.1 Graceful Graphs While graph theory is commonly considered to have begun in 1736, when Leonhard Euler solved and then generalized the famous Königsberg Bridge Problem, there was no real organization of this subject until 1936, when Dénes König wrote the first book [28] on the subject. Highlights of the first 200 years 1736–1936 of graph theory were described in a book by Norman Biggs, E. Keith Lloyd, and Robin Wilson [4] in 1976. The first two centuries of graph theory primarily consisted of some isolated results and contributions to recreational mathematics. Graph theory, however, never started developing into a theoretical and research area of mathematics until after World War II. This development essentially began during the 1950s and 1960s. The second book on the theory of graphs was written by the French mathematician Claude Berge [3] in 1958. The next such book was published in 1962 and written by Oystein Ore [34]. The book by Ore was the first written in English, as König’s was written in German and Berge’s in French. Late in the 1950s and well into the 1960s, conferences were organized whose major emphasis was on graph theory. One of these conferences, called the Symposium on the Theory of Graphs and Its Applications, took place in Smolenice © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6_2

7

8

2 Graceful Labelings

(in Czechoslovakia at the time) during June 17–20, 1963. The editors of the proceedings [14] of this conference wrote: It can easily be proved (a somewhat lengthy proof is given in this volume . . .) that the theory of graphs is growing steadily and has applications in so many fields of science as not many other branches have done before. A further important feature of graph theory is that a good part of its problems is understandable to a wide circle of readers. For this reason, for many years it has found its way into books on recreational mathematics with the object of attracting the interest of young students to mathematics. These facts led to the arrangement of Symposia on graph theory in Budapest 1958, Halle/Saale 1960 and, in 1963, in Smolenice.

The 1963 Symposium in Smolenice was quite likely the first major conference on graph theory. It featured 37 participants (24 from Czechoslovakia and 13 from outside the country). Among these participants were some of the best known mathematicians at that time known for their contributions to graph theory. This 4-day conference began with six half-day sessions, followed by discussions of problems on graph theory. The proceedings of this conference (see [14]) contained much of the mathematics presented in the lectures as well as 31 problems that were presented there. Problem #25, presented by Gerhard Ringel, stated a conjecture of his. Ringel’s Conjecture For every tree T of size m, the complete graph K2m+1 can be decomposed into 2m + 1 copies of T Later, Anton Kotzig presented an even stronger conjecture. Kotzig’s Conjecture For every tree T of size m, the complete graph K2m+1 can be cyclically decomposed into 2m + 1 copies of T We will return to these conjectures soon. Other conferences on the theory of graphs followed soon afterward, including one in Rome during July 5–9, 1966 and another in the United States in Kalamazoo, Michigan in 1968. Graph theory conferences became commonplace in the 1970s and thereafter. One of the participants of the Rome conference was Alexander Rosa who presented a paper [40] that would not only have a major influence on the area of graph labelings but would be directly related to the previously mentioned Ringel– Kotzig conjecture. In his paper, Rosa introduced four vertex labelings which he referred to as α- valuations, β-valuations, σ -valuations, and ρ-valuations. It was the β-valuation that would play a significant role in the study of labelings. This labeling would acquire a new name in 1972 when a β-valuation was referred to as a graceful labeling by Solomon Golomb [19]. A possible application of these labelings suggested by Golomb can be described as follows: Suppose that a connected graph of order n and size m represents a communications network having n terminals and m interconnections between terminals. Distinct identifying numbers are to be assigned to each terminal in a way that uniquely identifies the interconnections by assigning to each interconnection the absolute value of the difference of the numbers assigned to its two end-terminals. If the goal is to minimize the largest number assigned to a terminal, then the resulting problem is directly related to graceful labelings.

2.1 Graceful Graphs

9

Fig. 2.1 A graceful graph

Let G be a nonempty graph of order n and size m. The graph G has a graceful labeling f if the vertices of G can be assigned distinct elements from the set [0, m] = {0, 1, . . . , m} in such a way that the resulting edge labeling f  , defined by f  (uv) = |f (u) − f (v)| for every edge uv of G, results in distinct edges receiving distinct labels. That is, the function f : V (G) → [0, m] is injective and the function f  : E(G) → [m] = {1, 2, . . . , m} is bijective. If a graph G admits a graceful labeling, then G itself is called a graceful graph. For example, the graph G of order 5 and size 6 shown in Figure 2.1 is graceful. A graceful labeling f of G is also shown in Figure 2.1. Since |f (u) − f (v)| is the distance d (f (u), f (v)) between the integers f (u) and f (v) on the real line, another way to determine whether a given graph is graceful is suggested by this observation. Let G be a nonempty graph of order n and size m and consider the points 0, 1, 2, . . . , m − 1, m on the real line. The distance d (i, j) = |i − j| between any two of these points i, j ∈ [0, m], i = j, is one of the integers 1, 2, . . . , m. Suppose that there exists a subset V of [0, m] with |V | = n for which there is a set E of m pairs of distinct elements of V where the distances between the integers in these pairs are the elements of [m]. Let H be the graph with vertex set V and edge set E. If H ∼ = G, then G is graceful. If there is no such set V of n elements of [0, m] with this property, then G is not graceful. For example, consider the graph G shown in Figure 2.2. This graph G has order n = 6 and size m = 7. Consider the points 0, 1, . . . , 7 on the real line (also shown in Figure 2.2). Now consider the subset V = {0, 1, 2, 4, 6, 7} of the set {0, 1, . . . , 7} consisting of these six integers and the set E = {{1, 2}, {2, 4}, {1, 4}, {2, 6}, {1, 6}, {0, 6}, {0, 7}} of seven pairs of elements of V . The distances between the integers in these seven pairs are 1, 2, . . . , 7. The graph H with vertex set V and edge set E is isomorphic to G. Therefore, G is graceful. If f is a graceful labeling of a graph G of order n and size m, then the vertex labeling f of G defined by f (v) = m − f (v) for each vertex v of G

10

2 Graceful Labelings

Fig. 2.2 Showing that a graph G is graceful Fig. 2.3 The Petersen graph is graceful

is called the complementary labeling of f . The edge labeling f  defined by f  (uv) = |f (u) − f (v)| = |(m − f (u)) − (m − f (v))| = |f (u) − f (v)| = f  (uv) shows that f is also a graceful labeling of G. The complementary graceful labeling f of the graph G of Figure 2.1 is also shown in the figure. Another example of a graceful graph is the famous Petersen graph. A graceful labeling of this graph is shown in Figure 2.3. There is a slightly different but equivalent way to determine whether a graph G is graceful. Again, suppose that G is a graph of order n and size m. Consider the complete graph Km+1 whose m + 1 vertices are labeled 0, 1, . . . , m. If two vertices of Km+1 are labeled i and j with i > j, then the edge joining these two vertices is labeled i − j. The graph G is graceful if G is isomorphic to a subgraph of Km+1 whose m edges are labeled 1 through m in Km+1 . This is illustrated in Figure 2.4 for the graph G of Figure 2.1. In the vertex-labeled and edge-labeled complete graph Km+1 just described, there are m + 1 − i edges labeled i for 1 ≤ i ≤ m. Determining whether a given graph G of order n and size m is graceful by this method requires us to construct a subgraph

2.1 Graceful Graphs

11

Fig. 2.4 Showing that the graph G of Figure 2.1 is graceful

Fig. 2.5 The graphs C8 , C82 , and C83

of Km+1 that is isomorphic to G by selecting the single edge labeled m, one of the two edges labeled m − 1, one of the three edges labeled m − 2, and so on. There are other edge-labeled graphs of interest to us with edges labeled 1 through m, but where there are an equal number of edges of each label. Let G be a connected graph of order n. By the kth power G k of G is meant that graph where V (G k ) = V (G) such that uv ∈ E(G k ) if 1 ≤ dG (u, v) ≤ k, that is, if the distance between u and v is at most k in G. The graph G 2 is called the square of G and G 3 is called the cube of G. Observe for n ≥ 7 that the diameter of the n-cycle Cn is n/2 ≥ 3 and that for each such integer n, the graph Cn3 is 6-regular. More generally, for integers n and k with n ≥ 2k + 1 ≥ 5, diam(Cn ) = n/2 ≥ k and Cnk k is 2k-regular. In fact, if n = 2k + 1, then Cnk = C2k+1 = K2k+1 , which, of course, is 2k-regular. Furthermore, there are exactly n pairs of vertices in Cn whose distance in Cn is i where 1 ≤ i ≤ k. Each such pair of vertices in Cn is joined by an edge in Cnk . If we label each edge uv of Cnk by dCn (u, v), then Cnk contains exactly n edges labeled i for every integer i with 1 ≤ i ≤ k. This is illustrated in Figure 2.5 for n = 8 and k = 2, 3. The first graph is C8 , the second is its square C82 , and the third is its cube C83 in which each edge uv is labeled with dC8 (u, v). In Figure 2.5, a dashed line joins two vertices u and v with dC8 (u, v) = 2, while a bold line joins two vertices u and v with dC8 (u, v) = 3.

12

2 Graceful Labelings

2.2 Cyclic Decompositions One feature of graceful graphs G is that there is always a complete graph of a particular order that can be decomposed into copies of G—not only “decomposed” but “cyclically decomposed” into copies of G, which we are about to describe. Theorem 2.1 If G is a graceful graph of size m, then the complete graph K2m+1 can be cyclically decomposed into G. Proof Since G is graceful, there is a graceful labeling of G such that the vertices of G are labeled from a subset of {0, 1, . . . , m} so that the induced edge labels are 1, 2, . . . , m. Let V (K2m+1 ) = {v0 , v1 , . . . , v2m } where the vertices of K2m+1 are arranged cyclically in a regular (2m + 1)-gon and labeled in clockwise order. Denote the resulting (2m + 1)-cycle by C = (v0 , v1 , v2 , . . . , v2m−1 , v2m , v0 ). For each integer i with 0 ≤ i ≤ m, a vertex labeled i in the graceful labeling of G is placed at vi in K2m+1 . Every edge of G is drawn as a straight-line segment in K2m+1 , denoting the resulting copy of G in K2m+1 by G 1 . Hence, V (G 1 ) ⊆ {v0 , v1 , . . . , vm }. Each edge vs vt of K2m+1 (0 ≤ s, t ≤ 2m) is labeled dC (vs , vt ), where then 1 ≤ dC (vs , vt ) ≤ m. Consequently, K2m+1 contains exactly 2m + 1 edges labeled i for each i (1 ≤ i ≤ m) and G 1 contains exactly one edge labeled i for each integer i with 1 ≤ i ≤ m. Whenever an edge of G 1 is rotated clockwise through an angle of 2π k/(2m + 1) radians, where 1 ≤ k ≤ 2m, an edge of the same label is obtained. The subgraph obtained by rotating G 1 through a clockwise angle of 2π k/(2m + 1) radians is denoted by G k+1 . Then G k+1 ∼ = G and a cyclic decomposition of K2m+1 into 2m + 1 copies of G results.  Theorem 2.1 is illustrated for the graceful graph K3 in Figure 2.6, where a graceful labeling of K3 is shown. Since K3 has size m = 3 and 2m + 1 = 7, there is a cyclic decomposition of K7 into seven copies of K3 . The first copy G 1 of K3 is also shown in Figure 2.6 with solid lines. By rotating G 1 clockwise through an angle of 2π/7 radians, a second copy G 2 of K3 is obtained. The copy G 2 is also shown in Figure 2.6, where the edges of G 2 are drawn with dashed lines. Rotating G 1 five more times clockwise through angles of 2π/7 radians produces a cyclic decomposition of K7 into seven copies of K3 . Suppose that, instead of starting with the (2m + 1)-cycle C and the complete m , as described in the proof of Theorem 2.1, we were to begin graph K2m+1 = C2m+1 with a cycle Cp for some integer p ≥ 2m + 1 and consider the graph Cpm , where V (Cpm ) = {v0 , v1 , v2 , . . . , vp−1 }, whose vertices are arranged cyclically in a regular p-gon and labeled in clockwise order. If we proceed as in the proof but rotate G a total of p − 1 times, then we obtain the following. Theorem 2.2 Let G be a graceful graph of order n and size m and let p be an integer with p ≥ 2m + 1. Then the graph Cpm can be cyclically decomposed into p copies of G. Theorem 2.2 is illustrated for K3 and p = 9 in Figure 2.7.

2.2 Cyclic Decompositions

13

Fig. 2.6 Cyclically decomposing K7 into seven copies of K3

Fig. 2.7 Cyclically decomposing C93 into nine copies of K3

Of course, cyclically decomposing Cpm , where p ≥ 2m + 1, into p copies of a graph G of size m, according to Theorem 2.2, depends on G being graceful. Not all graphs are graceful, however. One such non-graceful graph is the 5-cycle C5 . Proposition 2.1 The 5-cycle C5 is not graceful. Proof Assume, to the contrary, that C5 is graceful. Since the size of C5 is 5, the five vertices of C5 must be labeled with five of the six labels 0, 1, . . . , 5 in such a way that the labels of the edges of C5 are 1, 2, 3, 4, 5. That is, exactly one of the six integers 0, 1, . . . , 5 is not a vertex label and exactly three edge labels are odd. Thus, either (1) three of the vertex labels are odd integers and two are even or (2) three of the vertex labels are even integers and two are odd. Since either the given graceful labeling or its complementary graceful labeling satisfies (1), we may assume that the given labeling satisfies (1). There are three odd labels and two even labels among the five edge labels. An edge uv has an odd label if and only if the labels of u and v are of opposite parity. Suppose that x and y are the two vertices of C5 having even labels. If x and y are adjacent on C5 , then only two edges of C5 have odd labels. If x and y are not adjacent, then four edges of C5 have odd labels. In either case, this is impossible.  Since C5 is not graceful, it therefore does not follow from Theorem 2.1 that the complete graph K11 can be cyclically decomposed into copies of C5 . Nevertheless, it can be so decomposed. The graph G 1 = C5 in Figure 2.8 has its edges

14

2 Graceful Labelings

Fig. 2.8 A cyclic decomposition of K11 into copies of C5

labeled 1, 2, 3, 4, 5, where an edge vs vt of G 1 (0 ≤ s, t ≤ 10) is labeled i ∈ [5] and dC11 (vs , vt ) = i. By rotating G 1 clockwise through an angle of 2π k/11 radians, where 1 ≤ k ≤ 10, another copy G k+1 of C5 is obtained where an edge labeled i in G 1 is rotated into another edge labeled i in G k+1 . Consequently, this results in a cyclic decomposition {G 1 , G 2 , . . . , G 11 } of K11 into 11 copies of C5 . We have seen for every graceful graph G of size m that there is a cyclic decomposition of K2m+1 into 2m + 1 copies of G. Yet, there may be a cyclic decomposition of K2m+1 into 2m + 1 copies of a non-graceful graph G of size m. Indeed, we just saw that the non-graceful graph C5 has this property. If a graph G can be drawn in the complete graph K2m+1 containing the Hamiltonian cycle C = (v0 , v1 , v2 , . . . , v2m−1 , v2m , v0 ) such that for each integer i with 1 ≤ i ≤ m, there is an edge xy of G such that dC (x, y) = i, then K2m+1 can be cyclically decomposed into 2m + 1 copies of G. If G can be drawn in the “first half” of K2m+1 , namely, in the complete subgraph Km+1 with vertex set {v0 , v1 , v2 , . . . , vm }, then G is graceful; while if this is not possible, then G is not graceful. This observation was also made by Rosa [40] in his original paper on the subject, dealing with another valuation he introduced. 5 5 = K11 . Not only can C11 be cyclically Since C11 has diameter 5, it follows that C11 5 5 decomposed into C5 , this is also the case for C13 and C15 , which is illustrated in Figure 2.9. This follows from three observations: (i) 5 − 2 + 3 + 1 + 4 = 11, (ii) 2 + 3 + 4 − 1 + 5 = 13, and (iii) 1 + 2 + 3 + 4 + 5 = 15. The so-called bow tie graph B = 2K2 ∨ K1 (the join of 2K2 and K1 ) shown in Figure 2.10 is another connected graph of order 5 that is not graceful. In fact, this graph B, the 5-cycle C5 , and the complete graph K5 are the only connected nongraceful graphs of order 5. Proposition 2.2 The bow tie graph B of Figure 2.10 is not graceful. Proof Assume, to the contrary, that B is graceful. Since the size of B is 6, the five vertices of B must be labeled with five of the seven labels 0, 1, . . . , 6 in such a way that the labels of the edges of B are 1, 2, . . . , 6. That is, the edges of B must be labeled with three even integers and three odd integers. If the three vertices of a triangle in

2.2 Cyclic Decompositions

15

5 and C 5 into copies of C Fig. 2.9 Illustrating cyclic decompositions of C13 5 15

Fig. 2.10 A cyclical decomposition of K13 into copies of the bow tie graph

B are all labeled with even integers or are all labeled with odd integers, then no edge of this triangle is labeled with an odd integer. If this is not the case, then only one vertex of a triangle is labeled with an even integer or only one vertex is labeled with an odd integer. In either case, exactly two edges of the triangles are labeled with an odd integer. Therefore, regardless of how the six vertices of B are labeled, an even number of edges of B are labeled with odd integers. This is impossible.  As with the graph C5 , even though B is not graceful, there is a cyclic decomposition of K13 into copies of B as shown in Figure 2.10. As we have seen, if G is a graceful graph of size m, then every graph Cpm where p ≥ 2m + 1 can be cyclically decomposed into copies of G. However, even if G is a non-graceful graph of size m, then it may occur that Cpm can be cyclically decomposed into copies of G for some integers p ≥ 2m + 1. For a given connected graph G of size m, graceful or not, there are numerous questions here regarding those integers p ≥ 2m + 1 for which the graph Cpm is decomposable into copies of G.

16

2 Graceful Labelings

Fig. 2.11 The six trees of order 6

The major question remaining here, however, is the following: W hich graphs are graceful? There are several well-known classes of graphs where it has been determined which members of the class are graceful. 1. 2. 3. 4.

The complete graph Kn (n ≥ 2) is graceful if and only if n ≤ 4 [19]. The cycle Cn is graceful if and only if n ≡ 0 (mod 4) or n ≡ 3 (mod 4) [40]. Every complete bipartite graph Kr,s is graceful [40]. Every wheel Wn = Cn−1 ∨ K1 is graceful [16].

Wheels were shown to be graceful by Roberto Frucht [16], who is well known for solving the problem introduced in König’s book [28] that for every finite group , there exists a graph whose automorphism group is isomorphic to . Paul Erd˝os and Robin Wilson [11] proved that almost all graphs are not graceful in 1977. That is, if G n is the set of all graphs of order n and Rn is the set of all graceful graphs of |Rn | = 0. order n, then limn→∞ |G n|

2.3 Graceful Trees In order for a graph G of order n and size m with n = m + 1 to be graceful, each of the integers 0, 1, 2, . . . , m must be used (exactly once) for a vertex label. If the graph G is connected, then G is a tree. Figure 2.11 shows all six trees of order 6. Each tree of order 6 is graceful. One way this can be verified is by beginning with the complete graph K6 whose vertices are labeled 0, 1, . . . , 5, where the edge joining the vertices labeled i and j (i > j) is itself labeled, with the label i − j. See Figure 2.12. Each of the trees T1 , T2 , . . . , T6 of order 6 can be found in K6 where the labels of the five edges in each tree are 1, 2, . . . , 5. See Figure 2.12. Therefore, all trees of order 6 are graceful. A double star is a tree of diameter 3. Thus, if T is a double star, then T contains exactly two vertices that are not end-vertices, called the central vertices of T . For integers s and t with s, t ≥ 2 and s + t = m + 1, let Ss,t denote the double star of size m whose central vertices have degrees s and t, respectively. Not only are the path P7 ,

2.3 Graceful Trees

17

Fig. 2.12 The graph K6 and the graceful trees T1 , T2 , . . . , T6

the star K1,6 , and the double star S3,4 of size 6 graceful, every path, star, and double star is graceful. Theorem 2.3 Every path, star, and double star is graceful. Proof First, we show that the path Pm+1 = (v0 , v1 , v2 , . . . , vm−1 , vm ) of size m ≥ 1 is graceful. Consider the following vertex labeling f of Pm+1 :  i if i is even 2 f (vi ) = if iis odd. m − i−1 2 This is a graceful labeling of Pm+1 . Consequently, every path is graceful. Next, we show that the star K1,m of size m ≥ 1 is graceful. Let v be the vertex of degree m in K1,m and let v1 , v2 , . . . , vm be the remaining vertices. Consider the following vertex labeling f of K1,m :

18

2 Graceful Labelings

Fig. 2.13 Graceful labelings of P7 , K1,6 , and S3,4

 f (x) =

0 if x = v i if x = vi for 1 ≤ i ≤ m.

This is a graceful labeling of K1,m . Therefore, every star is graceful. Finally, for integers s and t with s, t ≥ 2 and s + t = m + 1, we show that the double star Ss,t of size m ≥ 3 is graceful. Let u be the vertex of degree s and let v be the vertex of degree t in Ss,t . Let u1 , u2 , . . . , us−1 be the end-vertices adjacent to u and v1 , v2 , . . . , vt−1 the end-vertices adjacent to v. Consider the following vertex labeling f of Ss,t : ⎧ ⎪ ⎪ ⎨

0 m+1−i f (x) = ⎪ ⎪m + 1 − s ⎩ i

if x = u if x = ui for 1 ≤ i ≤ s − 1 ifx = v if x = vi for 1 ≤ i ≤ t − 1.

Since this labeling is graceful, every double star is graceful.



Figure 2.13 illustrates the graceful labelings for the path P7 , star K1,6 , and double star S3,4 , as described in Theorem 2.3. Indeed, it is believed that every nontrivial tree is graceful. This is the most famous conjecture in the area of graceful labelings. The Graceful Tree Conjecture Every nontrivial tree is graceful. The Graceful Tree Conjecture is credited jointly to Anton Kotzig (Rosa’s advisor) and Gerhard Ringel [38], who with J. W. T. Youngs is known for completing the solution of the famous Heawood Map Coloring Problem [39], namely, the problem of determining the largest chromatic number of a graph that can be embedded on an orientable surface of positive genus (see [9, p. 219]). The Graceful Tree Conjecture has been verified for several classes of trees, including the following:

2.3 Graceful Trees

1. 2. 3. 4.

19

Every caterpillar is graceful [40]. Every tree with at most four end-vertices is graceful [24]. Every tree of order at most 35 is graceful [12, 18]. Every tree of diameter at most 5 is graceful [23].

Joseph Gallian’s dynamic survey [18] of graph labelings contains a vast list of periodically updated results on the subject.

Chapter 3

Harmonious Labelings

In this chapter, we discuss vertex labelings where each label is selected from Zm for a graph of size m. This labeling then gives rise to an edge labeling where the label of an edge is the sum of the labels of its incident vertices. The primary interest is when the edge labels are distinct.

3.1 Harmonious Graphs Thering  Z3 of integers modulo 3 consists of the three elements 0, 1, 2. When adding the 23 = 3 pairs of elements of Z3 , we obtain all elements of Z3 : 0 + 1 = 1, 0 + 2 = 2, 1 + 2 = 0. The ring Z6 consists of the six elements 0, 1, 2, 3, 4, 5. It turns out that there exists a set S of four elements of Z6 such that when the 24 = 6 pairs of elements of S are added, the six resulting sums are precisely the elements of Z6 . In particular, for S = {0, 1, 2, 4}, we have 0 + 1 = 1, 0 + 2 = 2, 0 + 4 = 4, 1 + 2 = 3, 1 + 4 = 5, 2 + 4 = 0. This set S consisting of four elements of Z6 is referred to as an additive basis for Z6 in that every   element of Z6 can be expressed as the sum of two distinct elements of S. Since 25 = 10, this brings up the question as to whether Z10 has an additive basis consisting of five elements of Z10 . The answer to this question is no. Indeed,  for every integer m = n2 , where n ≥ 5, there is no additive basis for Zm consisting of n elements of Zm . The observations mentioned above suggest a vertex labeling concept. For example, consider the complete graph K 3 whose vertices are labeled with the elements 0, 1, 2 of Z3 . From this vertex labeling, an edge labeling of K 3 is produced where the label of an edge is the sum (in Z3 ) of the labels of its two incident vertices and the three edge © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6_3

21

22

3 Harmonious Labelings

Fig. 3.1 Harmonious labelings of K 3 and K 4

labels are distinct. This is shown in Figure 3.1. In the same figure, K 4 is also shown where the four vertices are labeled with the distinct elements of S = {0, 1, 2, 4} of Z6 . As we already noted, there is no way to label the five vertices of K 5 in such a way that the resulting labels of the ten edges of K 5 are the ten elements 0, 1, . . . , 9 of Z10 . More generally, let G be a connected graph of order n and size m. Even though m ≥ n − 1, let us assume for the present that m ≥ n. The graph G is called a harmonious graph if there exists a set S of n distinct elements of Zm with the property that the elements of S can be assigned to the vertices of G in such a way that each edge uv of G is labeled with the sum in Zm of the labels of u and v and the resulting edge labels are distinct, namely, 0, 1, . . . , m − 1. Such a labeling of G is called a harmonious labeling. Hence, for a connected graph G of order n and size m with m ≥ n, a vertex labeling f of G is a harmonious labeling if f : V (G) → Zm is injective and the resulting edge labeling f  : E(G) → Zm defined by f (uv) = f (u) + f (v) is bijective. A connected graph is therefore harmonious if it admits a harmonious labeling. Thus, K 3 and K 4 are harmonious, while for each integer n ≥ 5, the graph K n is not. Since a connected graph of order n and size m with m = n − 1 is a tree, the condition m ≥ n restricts ourselves to connected graphs that are not trees. The topic of harmonious trees will be discussed in the next section. Consequently, to determine whether a connected graph of order n and size m with m ≥ n is harmonious, it is necessary to determine whether there exists a harmonious labeling of G, that is, whether the vertices of G can be labeled with n distinct elements of Zm such that the resulting edge labels of G are the elements of Zm . For example, for the famous Petersen graph, which is a cubic graph of order 10 and size 15, there are 10 distinct elements of Z15 that can be assigned to the vertices of the Petersen graph so that the resulting edge labels are the elements of Z15 . A harmonious labeling of the Petersen graph is shown in Figure 3.2. Additive bases and the resulting concept of harmonious labelings of graphs were introduced by Ronald Graham and Neil Sloane [20] in 1980, who, at that time, were members of Bell Laboratories. Graham is known for his many contributions to discrete mathematics and Sloane is probably best known for introducing the Online Encyclopedia of Integer Sequences. Let m ≥ 3 be an integer and consider a regular m-gon whose m vertices are labeled cyclically in clockwise order with the elements 0, 1, . . . , m − 1 of Zm . Every two

3.1 Harmonious Graphs

23

Fig. 3.2 The Petersen graph is harmonious

Fig. 3.3 The complete graph K m labeled with elements of Zm for m = 6 and m = 7

vertices of this m-gon are then joined by a straight-line segment. For two vertices of the m-gon labeled i and j, the edge joining them is labeled i + j = k ∈ Zm . All edges in this complete graph K m that are parallel to an edge labeled k are also labeled k. See Figure 3.3 for the complete graphs K 6 and K 7 . Let G be a connected graph of order n and size m with m ≥ n. If the vertices and edges of K m are labeled with the elements of Zm as indicated above and there exists one edge of each parallel class that results in a graph isomorphic to G, then G is harmonious. Figure 3.4 shows three different harmonious graphs of size 6, no two of which have the same order. We have mentioned that only the complete graphs K n with n ≤ 4 are harmonious, which is exactly the same situation for graceful complete graphs. We also saw in Chapter 2 that “half” of the cycles are graceful. This too is true for harmonious cycles, although it is not the same half. Theorem 3.1 The cycle Cn , n ≥ 3, is harmonious if and only if n is odd.

24

3 Harmonious Labelings

Fig. 3.4 Three harmonious graphs of size 6 having different orders

Fig. 3.5 Harmonious labelings of C5 and C7

Proof We have already seen that C3 = K 3 is harmonious. For an odd integer n ≥ 5, let Cn = (v0 , v1 , . . . , vn−1 , v0 ), where the vertices of Cn are listed in clockwise order. We now label the vertex vi , 0 ≤ i ≤ n − 1, with the element i ∈ Zn (see Figure 3.5). Beginning with the edge v0 v1 and proceeding clockwise about Cn , the edge labels are then 1, 3, 5, . . . , n − 2, 0, 2, 4, . . . , n − 1. This is a harmonious labeling of Cn . Next, let n ≥ 4 be an even integer. Assume, to the contrary, that there is a harmonious labeling of Cn , where, say, vi is labeled ai ∈ Zn for 0 ≤ i ≤ n − 1. Hence, Zn = {a0 , a1 , . . . , an−1 }. Thus, n−1  i=0

ai =

n−1  i=0

i=

  n (mod n). 2

Since each vertex of Cn contributes its label as a term in the sum of two edge labels, it follows that     n−1 n−1   n n =2 ai = 2 i =2 (mod n). 2 2 i=0 i=0    Consequently, 2 n2 ≡ n2 (mod n) and so n2 ≡ 0 (mod n). However then, n | n(n−1) , which is impossible since (n − 1)/2 is not an integer.  2

3.2 Harmonious Trees

25

3.2 Harmonious Trees As we mentioned earlier, we are interested in those connected graphs of order n and size m for which a harmonious labeling is possible. Thus far, we have only discussed connected graphs with m ≥ n. However, it is possible that m = n − 1 for a connected graph G, in which case G is a tree. It is harmonious trees that we discuss now. Since the labels used in a harmonious labeling of a tree T of order n and size m = n − 1 are the elements of Zm , it is impossible to assign n distinct elements of Zm to the n vertices of T . To rectify this situation in their definition of harmonious labeling, Graham and Sloane allowed exactly two vertices of a tree to be assigned the same label. It is easy to see that all paths and stars are harmonious. Indeed, a harmonious labeling of a path of even order is much like that of a cycle. For paths Pn = (v0 , v1 , . . . , vn−1 ) of odd order n = 2k + 1 and size n − 1 = 2k, the vertices are labeled so that the induced edge labels are f  (vi vi+1 ) = i + k for 0 ≤ i ≤ k − 1 and f  (vi vi+1 ) = i − k for k ≤ i ≤ 2k − 1. By defining the label f (v0 ) = 0, the labels f (vi ) of the remaining vertices vi , 1 ≤ i ≤ n − 1, are all uniquely determined. Furthermore, all vertex labels are distinct except that f (v1 ) = f (vn−1 ) = k. Harmonious labelings of Pn for n = 6, 7, 8, 9 and the star K 1,6 are shown in Figure 3.6. Many other classes of trees have been shown to be harmonious. Among them are the following: 1. Every caterpillar is harmonious [20]. 2. Every tree of order at most 31 is harmonious [12, 18]. In fact, Graham and Sloane [20] made the following conjecture, which parallels the famous conjecture on graceful labelings of trees. The Harmonious Tree Conjecture Every nontrivial tree is harmonious. If f is a harmonious labeling of a connected graph G of size m, then the labeling f : V (G) → Zm of G defined by f (v) = m − 1 − f (v) for each v ∈ V (G) is called

Fig. 3.6 Harmonious labelings of Pn for n = 6, 7, 8, 9 and K 1,6

26

3 Harmonious Labelings

the complementary labeling of f and is also a harmonious labeling of G. If T is a nontrivial tree of size m and the label i ∈ Zm is repeated in a harmonious labeling f of T , then m − 1 − i is the repeated label in f . If m is odd and i = (m − 1)/2, then the repeated label is the same in both f and f . For a nontrivial tree T of size m, let h(T ) denote the number of elements of Zm that can be repeated in some harmonious labeling of T . Therefore, 0 ≤ h(T ) ≤ m for every tree T of size m. Of course, if the Harmonious Tree Conjecture is true, then 1 ≤ h(T ) ≤ m. In fact, if T is a tree of even size m and the Harmonious Tree Conjecture is true, then h(T ) is even and 2 ≤ h(T ) ≤ m. Figure 3.7 shows four trees, each of size m where m is 2, 3, or 5. For each of these four trees T , it follows that h(T ) = m. That is, every element of Zm is the repeated label in some harmonious labeling of T . This brings up several questions. Problem 3.1 Does there exist a class of trees T for which h(T ) = 1? Problem 3.2 For which pairs k, m of positive integers with k ≤ m, does there exist a tree T of size m for which h(T ) = k? We mentioned earlier that to determine whether a connected graph G of order n and size m with m ≥ n is harmonious, we can begin with a regular m-gon whose vertices are labeled cyclically in clockwise order with the elements 0, 1, 2, . . . , m − 1 of Zm . Every two vertices of this m-gon, labeled by i and j say, are joined by a straight-line segment labeled i + j ∈ Zm . This produces the complete graph K m , each of whose

Fig. 3.7 Trees T for which h(T ) = |E(T )|

3.2 Harmonious Trees

27

edges is labeled with an element of Zm . Then two edges of K m have the same label if and only if they are parallel. Thus, the edge set of K m is partitioned into m parallel classes. To determine whether the graph G is harmonious, we then need to determine if one edge can be selected from each of the m parallel classes to arrive at a graph isomorphic to G. There is only one harmonious graph of size 2, namely, the path P3 (see Figure 3.8). There are two harmonious trees of size 3, namely, the path P4 and the star K 1,3 . There is only one other harmonious graph of size 3. If we begin with a regular 3-gon, then there is one edge in each parallel class, that is, K 3 is the remaining harmonious graph of size 3. There are three trees of size 4, all harmonious. To determine which connected graphs of size 4 that are not trees but are also harmonious, we begin with a regular 4-gon, whose vertices are labeled cyclically with the elements 0, 1, 2, 3 of Z4 in clockwise order (see Figure 3.8). There is only one edge in the parallel class joining the vertices labeled 0 and 2 and one edge in the parallel class joining the vertices labeled 1 and 3. So, these two edges must be selected. There are two edges in the parallel class joining the vertices labeled 0 and 1 (as well as in the parallel class joining the vertices labeled 1 and 2). Only one harmonious graph results here, shown in Figure 3.8. In addition to the six trees of size 5, the harmonious graphs of size 5 consist of the four graphs shown in Figure 3.9. Each harmonious graph of size m ≥ 6 that is not a tree can be found in a similar manner, namely, by beginning with a regular m-gon whose vertices are labeled cyclically in clockwise order with the elements 0, 1, . . . , m − 1 of Zm and selecting one edge from each of the m parallel classes. For m = 6, one harmonious graph obtained is G = (K 4 − e) + K 2 , which is disconnected. A harmonious labeling f : V (G) → Z6 of G can be defined by assigning the labels 1 and 4 to the two vertices of degree 3

Fig. 3.8 Harmonious graphs of sizes 2, 3, and 4

28

3 Harmonious Labelings

Fig. 3.9 Harmonious graphs of size 5 that are not trees

in G, the labels 0 and 2 to the two vertices of degree 2 in G, and the labels 3 and 5 to the two end-vertices in G. This harmonious graph has two components. This brings up the problem of determining a harmonious disconnected graph with a specified number of components.

Chapter 4

Prime Labelings

Unlike the situation with graceful labelings and harmonious labelings, where a vertex labeling induces an edge labeling, the emphasis in this chapter is on graphs whose vertices can be labeled with positive integers so that the labels of every two adjacent vertices have a particular number-theoretic property. We also see that some of these labelings can be looked at in a set-theoretic manner.

4.1 Prime Graphs Each of the vertex labelings described in the two preceding chapters has led to an edge labeling possessing a property of interest. In this chapter, once again our interest lies with graphs possessing certain vertex labelings, but in many instances there is no corresponding edge labeling. The labelings we present in this chapter have either a number-theoretic or a set-theoretic flavor to them. In each instance, however, prime numbers play a role, either directly or indirectly. Let G be a graph of order n. By a prime labeling of G is meant a vertex labeling of G with the distinct integers in the set [n] = {1, 2, . . . , n} such that the labels of every two adjacent vertices of G are relatively prime (also called coprime). That is, a prime labeling is a bijective function f : V (G) → [n] such that gcd( f (u), f (v)) = 1 for each pair u, v of adjacent vertices of G. If there exists a prime labeling of G, then G is called a prime graph. This concept was originated by Roger Entringer (see [18]) around 1980. For example, the graph Q 3 of the cube is a prime graph. A prime labeling of this graph is shown in Figure 4.1. While adjacent vertices must have relatively prime labels, nonadjacent vertices may have relatively prime labels as well. There are certain classes of graphs, the prime graphs of which can be determined quite quickly. One of these is the class of cycles. Observation 4.1 Every cycle is a prime graph. Proof Let Cn = (v1 , v2 , . . . , vn , v1 ), where n ≥ 3. Define the labeling f : V (Cn ) → [n] by f (vi ) = i for all i (1 ≤ i ≤ n). Since every pair of consecutive positive inte© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6_4

29

30

4 Prime Labelings

Fig. 4.1 A prime graph Q 3

gers are relatively prime (as are 1 and every other positive integer), this is a prime labeling of Cn .   By Observation 4.1, C3 = K 3 is therefore a prime graph. However, K 4 is not prime. Regardless of how the vertices of K 4 are labeled with distinct elements of [4], the two vertices labeled 2 and 4 cannot be adjacent. Similarly, K n is not prime for any integer n ≥ 4. Indeed, this observation tells us that for each integer n ≥ 4, graphs of order n and sufficiently large size cannot be prime. That is, the smaller the size of a graph, the more likely it is to be prime. Since the 4-cycle C4 is prime and C4 = K 2,2 , the complete bipartite graph K 2,2 is prime. In fact, all complete bipartite graphs K 2,t where t ≥ 2 are prime. To see this, it is helpful to make use of a well-known theorem from number theory. Bertrand’s Postulate For every integer n ≥ 2, there is at least one prime p such that n < p < 2n. This theorem is named for Joseph Bertrand, a French mathematician who conjectured in 1845 (in his early 20s) that for every integer n ≥ 4, there is at least one prime p with n < p < 2n − 2 (or for n ≥ 2, there is at least one prime p with n < p < 2n). The Russian mathematician Pafnuty Chebyshev verified this conjecture in 1850, when the resulting theorem became known as Bertrand’s Postulate. In 1919, the famous Indian mathematician Srinivasa Ramanujan gave a shorter proof of Bertrand’s Postulate. In 1932, the 19-year-old Hungarian mathematician Paul Erd˝os presented (in his first paper) an elementary proof of this fact. Even though Chebyshev initially proved this result, Erd˝os’s proof was universally considered more elegant. What Erd˝os had accomplished became well known among Hungarian mathematicians and the news of what he had done was often accompanied by the rhyme: Chebyshev said it, And I say it again. There is always a prime Between n and 2n. Now, we show that the graph K 2,t is prime for every integer t ≥ 2. Theorem 4.2 For each integer t ≥ 2, the complete bipartite graph K 2,t is prime.

4.1 Prime Graphs

31

5

1

2

3

4

7

6

8

9

Fig. 4.2 A prime labeling of K 3,6

Proof Let U and W be the partite sets of K 2,t where |U | = 2 and |W | = t. The labels 1 and the largest prime p ∈ [t + 2] are assigned to the two vertices of U and the remaining integers in the set S = [t + 2] − {1, p} are assigned to the vertices of W . Surely, 1 and every integer in S are relatively prime. Also, p and every integer k ∈ S with k < p are relatively prime. We claim that p and every integer k ∈ S with k > p are also relatively prime. If this were not the case, then S must contain an integer q > p such that p | q and so q ≥ 2 p. Since q ∈ S, it follows that 2 p ∈ S. By Bertrand’s Postulate, there is a prime p  such that p < p  < 2 p. But then p  ∈ S   and p  > p, which is impossible. Thus, we obtain a prime labeling of K 2,t . Theorem 4.2 then brings up the question of the primality of the complete bipartite graphs K 3,t where t ≥ 3. First, it can be immediately shown that not all such graphs are prime, as K 3,3 itself is not prime. Not only is the graph K 3,3 not prime, but for every integer r ≥ 3, the graph K r,r is not prime. Theorem 4.3 For each integer r ≥ 3, the graph K r,r is not prime. Proof Suppose for some integer r ≥ 3 that K r,r is prime. Let U and W be the partite sets of K r,r . Thus, |U | = |W | = r . Since half of the integers in [2r ] are even, all r of these even integers must be labels of vertices in the same partite set, say U . That is, every vertex of U is labeled with an even integer. Since r ≥ 3, the vertices labeled 3 and 6 must be in the same partite set. Since the vertex labeled 6 is in U , the vertex labeled 3 is also in U , which is impossible.   Even though K 3,3 is not prime, all three graphs K 3,4 , K 3,5 , and K 3,6 are prime. Assigning the labels 1, 5, 7 to the three vertices in the partite set of size 3 in each of these graphs produces a prime labeling of the graph. A prime labeling of K 3,6 is shown in Figure 4.2. Even though K 3,4 , K 3,5 , and K 3,6 are prime, the graph K 3,7 is not. Proposition 4.1 The complete bipartite graph K 3,7 is not prime. Proof Suppose that there is a prime labeling of K 3,7 with the integers in the set [10]. Let U and W be the partite sets of K 3,7 where |U | = 3 and |W | = 7. Since no two

32

4 Prime Labelings

Fig. 4.3 All trees of order 6 are prime

of the integers 2, 4, 6, 8, and 10 are relatively prime, these integers must be assigned to vertices in the same partite set, necessarily to five of the seven vertices of W . However, no two of the integers 3, 6, and 9 are relatively prime and 5 and 10 are not relatively prime. So, 3, 9, and 5 must also be assigned to vertices of W . This leaves only the integers 1 and 7 as the possible labels of the three vertices of U , which is impossible.   While K 3,7 is not prime, it has been shown by M. A. Seoud, A. T. Diab, and E. A. Elsakhawi [43] that K 3,t is prime, however, for each integer t ≥ 8. The graph G shown in Figure 4.3 is the prime graph of order 6 having maximum size. Therefore, any prime graph of order 6 must be a spanning subgraph of G. As Figure 4.3 also shows, every tree of order 6 is therefore prime, which brings up the question as to which trees are prime (or perhaps not prime). Trees belonging to certain classes have been shown to be prime, including the following. 1. All paths and stars are prime [17]. 2. All trees of order at most 50 are prime [18]. 3. All caterpillars with maximum degree at most 5 are prime [50]. In fact, Roger Entringer (see [18, 50]) made the following conjecture in the 1980s. The Prime Tree Conjecture Every nontrivial tree is prime. In 2011, Penny Haxell, Oleg Pikhurko, and Anusch Taraz [22] proved that all trees having a sufficiently large order are prime.

4.1 Prime Graphs

33

Fig. 4.4 Graphs of order n and size 2 n/2 + 1 that are not prime

Theorem 4.4 There exists a positive integer n  such that for each integer n > n  , every tree of order n is prime. The Prime Tree Conjecture remains open, however. Since, by Observation 4.1, every cycle is prime and every tree is conjectured to be prime, it is not surprising that M.A. Seoud and M. Z. Youssef [45] made the following conjecture. A unicyclic graph is a connected graph with exactly one cycle. Thus, the order and size of a unicyclic graph are the same. Conjecture 4.1 Every unicyclic graph is prime. From what we’ve seen then, every connected graph of order n whose size is either n − 1 or n is either prime or is conjectured to be prime. In [22], Haxell, Pikhurko, and Taraz observed that for each integer n ≥ 8, there exists a graph of order n and size 2 n/2 + 1 that is not prime. The graphs G 1 and G 2 of Figure 4.4 have this property. For example, if there were a prime labeling of G 1 with integers in the set [8], then at most one vertex among u 1 , u 2 , u 3 can have an even label, at most one of u 4 , u 5 can have an even label and at most one of u 6 , u 7 , u 8 can have an even label, which is impossible. In the same paper [22], the following was verified, thereby establishing the truth of an earlier conjecture by S. N. Rao [37]. Theorem 4.5 There exists a positive integer n  such that for each integer n > n  , the minimum size of a non-prime graph of order n is 2 n/2 + 1. Since the size of a unicyclic graph equals its order, it follows by Theorem 4.5 that Conjecture 4.1 is true for unicyclic graphs of sufficiently large even order. If a graph G of order n is not prime, then, by definition, there is no prime labeling of G with labels in the set [n]. However, there is always a sufficiently large integer k ≥ n and a vertex labeling of G using n distinct integers in the set [k] such that the labels of every two adjacent vertices of G are relatively prime. Such a labeling is often called a coprime labeling of G. The smallest such integer k that accomplishes this is called the coprime index of G, denoted by cp(G). For example, if we assign the labels 1, 3, 5 to one partite set of K 3,3 and the labels 2, 4, 7 to the other partite set, then we obtain a coprime labeling of K 3,3 using six integers in the set [7]. Hence, cp(K 3,3 ) = 7. Similarly, we can assign the labels 1, 3, 5, 9 to one partite set of K 4,4 and the labels 2, 4, 7, 8 to the other partite set to obtain a coprime labeling of K 4,4 using eight integers in the set [9]. Thus, cp(K 4,4 ) = 9. The values of cp(K r,r )

34

4 Prime Labelings

Fig. 4.5 gcd-rainbow labelings of K 3 , C5 , and P6

(3 ≤ r ≤ 13) were determined in [46]. The values of cp(K r,r ) have been determined for even larger values of r . r 3 4 5 6 7 8 9 10 11 12 13 cp(K r,r ) 7 9 11 15 17 21 23 27 29 32 27

There is a concept related to the coprime index. Suppose that G is a graph of order n with coprime index k. Then there exists a vertex labeling  of G with n distinct elements of the set [k] where every edge uv of G is assigned the color gcd((u), (v)) = 1, that is, there exists a monochromatic edge coloring of G, where every edge of G is colored 1. Another well-studied edge coloring of a graph is that of a rainbow coloring, where no two edges of the graph are colored the same. For a nonempty graph G of order n and an integer k ≥ n, a gcd-rainbow labeling of G is a vertex labeling  with n distinct integers in the set [2, k] such that each edge uv of G is assigned the color gcd((u), (v)) and all edges of G have distinct colors. The gcd-rainbow index gr(G) is the smallest integer k ≥ n + 1 such that G has a gcd-rainbow labeling with n elements of [2, k]. For example, gr(K 3 ) = 6, gr(C5 ) = 12, and gr(P6 ) = 10. See Figure 4.5. That the gcd-rainbow index exists for every nonempty graph is established next. Theorem 4.6 Every nonempty graph has a gcd-rainbow labeling. Proof Let G = K n for n ≥ 2 with V (G) = {v1 , v2 , . . . ,vn } and let S = { p1 , p2 , . . . , pn } n = n distinct (n − 1)be the set consisting of the first n primes. Then there are n−1 element subsets S1 , S2 , . . . , Sn of S. For i = 1, 2, . . . , n, the vertex vi is assigned the label which is the product of the elements of Si . Therefore, the vertices of G have distinct labels. For vertices u and v of G, the edge uv is assigned the color which is the greatest common divisor of the labels of uandv. Thus,   the colors of the edges n = n2 distinct (n − 2)-element of G are the products of the elements in the n−2 subsets of S. Thus, this is a gcd-rainbow labeling of G. Since any graph H of order n is a spanning subgraph of G, the restriction of a gcd-rainbow labeling of G to H is a gcd-rainbow labeling of H . Therefore, every nonempty graph has a gcd-rainbow labeling.  

4.1 Prime Graphs

35

n The proof of Theorem 4.6 shows for a graph G of order n that gr(G) ≤ i=2 pi , where p1 , p2 , . . . , pn are the first n primes. In particular, gr(K 3 ) ≤ 15. However, we have already seen that gr(K 3 ) = 6. For all three graphs shown in Figure 4.5, the edge colors are the elements of the set [m], where m is the size of the graph. Among the questions involving this concept is therefore the following. Problem 4.1 For which graphs G of size m is there a gcd-rainbow labeling of G whose set of edge colors is [m]?

4.2 Multi-prime Labelings We now describe another vertex labeling of graphs where, once again, adjacent vertices are labeled with relatively prime integers. A vertex labeling f of a nonempty connected graph G is a multi-prime labeling if f : V (G) → [2, ∞), where uv ∈ E(G) if and only if gcd( f (u), f (v)) = 1. There are some major differences between a multi-prime labeling and a prime (or coprime) labeling. First, rather than the vertex labels coming from a set [n] (or [k]), they come from the set [2, ∞). Second, a multi-prime labeling is not required to be injective. Third, adjacent vertices are the only pairs of vertices whose labels are relatively prime. Thus, in a multi-prime labeling, 1 is not the label of any vertex and so every label is divisible by at least one prime. Furthermore, if u and v are two nonadjacent vertices, then their labels are not relatively prime and so there is at least one prime that divides both labels. There is no requirement in a multi-prime labeling of a graph that distinct vertices must be assigned distinct labels. If two vertices are labeled the same, however, then these vertices cannot be adjacent. A multi-prime labeling of the complete graph K n of order n ≥ 3 can be obtained by assigning distinct primes to distinct vertices. Thus, the vertex labeling of C3 shown in Figure 4.6 is a multi-prime labeling. Figure 4.6 also shows a multi-prime labeling of C4 = K 2,2 . In fact, a multi-prime labeling of a complete multipartite graph G can be obtained by assigning the same prime to two vertices of G if and only if these two vertices belong to the same partite set. Next, we consider the 5-cycle C5 . A multi-prime labeling of C5 is shown in Figure 4.7. In this labeling, each label is divisible by at least one of the five primes 2, 3, 5, 7, 11 (actually by exactly two of these primes). One might ask if there is a multi-prime labeling f of C5 where fewer primes are used. First, we observe that

Fig. 4.6 Multi-prime labelings of C3 and C4

36

4 Prime Labelings

Fig. 4.7 A multi-prime labeling of C5

no label in a multi-prime labeling can be a prime. Suppose, say, that f (v1 ) = p is a prime. In this case, since v1 v3 and v1 v4 are not edges of C5 , it follows that p | f (v3 ) and p | f (v4 ), which is impossible since v3 and v4 are adjacent, which requires f (v3 ) and f (v4 ) to be relatively prime. Thus, every label must be divisible by at least two distinct primes and the labels of two adjacent vertices must be divisible by at least one of four distinct primes, say 2, 3, 5, 7. Suppose then that each vertex label of C5 is divisible by one of the four primes 2, 3, 5, 7. No vertex label can be divisible by three of these primes, for otherwise, the label of each neighbor of this vertex is divisible by a single prime. Therefore, each vertex label is divisible by exactly two primes, say f (v1 ) = 2 · 3 and f (v2 ) = 5 · 7. We may assume that 2 | f (v3 ) and 3 | f (v4 ). Also, we may assume that 5 | f (v4 ) and 7 | f (v5 ). However then, 2 | f (v5 ), which is impossible since v1 and v5 are adjacent and 2 | f (v1 ). Therefore, every multi-prime labeling of C5 requires the use of at least five primes. Thus, five primes is the smallest number of primes that can be used in a multi-prime labeling of C5 . This example brings up a parameter associated with multi-prime labelings of graphs. For a nonempty connected graph G, the multi-prime index ρ(G) is the minimum number of primes p in a multi-prime labeling f of G such that p | f (x) for at least one vertex x of G. Hence, ρ(G) ≥ 2 for every nonempty connected graph G. Thus, ρ(K n ) = n and, as we have observed, ρ(C5 ) = 5. The following observation is often useful. Observation 4.7 If H is an induced subgraph of a graph G, then ρ(H ) ≤ ρ(G). For a given nontrivial connected graph G, there is a class of graphs associated with G, all of which have the same multi-prime index as G. To see this, let V (G) = {v1 , v2 , . . . , vn } and let H be the graph obtained from G by replacing each vertex vi (1 ≤ i ≤ n) of G with the empty graph K qi of order qi . Hence, the vertex set of n V (K qi ) and two vertices u and w of H are adjacent in H if u ∈ V (K qi ) H is ∪i=1 and w ∈ V (K q j ) where vi v j ∈ E(G). The graph H is referred to as the composition graph of G and K q1 , K q2 , . . . , K qn and is often denoted by G[K q1 , K q2 , . . . , K qn ]. Proposition 4.2 For a nontrivial connected graph G with V (G) = {v1 , v2 , . . . , vn }, let H be the composition graph of G and K q1 , K q2 , . . . , K qn . Then ρ(H ) = ρ(G). Proof Suppose that ρ(G) = k. For each integer i with 1 ≤ i ≤ n, let wi ∈ V (K qi ) and let S = {w1 , w2 , . . . , wn }. Since H [S] ∼ = G and H [S] is an induced subgraph

4.2 Multi-prime Labelings

37

of H , it follows by Observation 4.7 that k = ρ(G) = ρ(H [S]) ≤ ρ(H ). Next, let there be given a multi-prime labeling f : V (G) → [2, ∞) of G. Define the labeling g : V (H ) → [2, ∞) by g(u) = f (vi ) if u ∈ V (K qi ) where 1 ≤ i ≤ n. Since g is a multi-prime labeling of H , it follows that ρ(H ) ≤ k = ρ(G). Therefore, ρ(H ) = ρ(G) = k.   Since a complete k-partite graph G is the composition graph of K k and k empty graphs, the following is an immediate consequence of Proposition 4.2. Corollary 4.1 If G is a complete k-partite graph, k ≥ 2, then ρ(G) = ρ(K k ) = k. As noted earlier, in a multi-prime labeling of a graph, the vertex labels are not required to be distinct. However, if two labels are the same, such as can occur with vertices belonging to the same partite set of a complete multipartite graph, primes can be raised to higher powers to produce a multi-prime labeling with distinct labels without increasing the number of primes used. For example, if a multi-prime labeling of a complete multipartite graph G assigns a prime p to every vertex in a partite set of G and the partite set consists of r ≥ 2 vertices, then p, p 2 , . . . , pr may be assigned to these vertices to produce distinct labels. This, however, has no effect on the value of ρ(G). Unlike the situation for prime labelings, where a graph may or may not have such a labeling, every nontrivial connected graph has a multi-prime labeling. Let P = { p1 , p2 , p3 , . . .} be the set of distinct primes, where we can assume that p1 = 2 < p2 = 3 < p3 = 5 < . . . and so on. Theorem 4.8 Every nontrivial connected graph has a multi-prime labeling. Proof We proceed by induction on the order n of a connected graph. The result is immediate for small values of n, say n = 2, 3, 4. Assume that the statement is true for all connected graphs of order n for an integer n ≥ 4. Let G be a connected graph of order n + 1, let v be a non-cut-vertex of G, and let G  = G − v. Since G  is a connected graph of order n, it follows by the induction hypothesis that G  has a multi-prime labeling. Let such a labeling f  of G  be given using k primes, say p1 , p2 , . . . , pk . Suppose that V (G  ) = {u 1 , u 2 , . . . , u n }, where N (v) = {u 1 , u 2 , . . . , u r }, 1 ≤ r ≤ n. Define a vertex labeling f of G by  f (x) =

if x = u i for 1 ≤ i ≤ n pi+k f  (u i ) k+n pn+k+1 i=k+r +1 pi if x = v.

Since x y ∈ E(G) if and only if gcd( f (x), f (y)) = 1, it follows that f is a multiprime labeling of G.  

38

4 Prime Labelings

4.3 Subset Labelings The multi-prime labelings encountered in the preceding section can be looked at in another way. First, let us review the concept of a multi-prime labeling f of a graph G. Once again, let 2 = p1 , 3 = p2 , 5 = p3 , p4 , p5 , . . . be the primes. Then for two vertices u and v of G, it follows that u and v are assigned labels f (u) = pi1 pi2 · · · pik and f (v) = p j1 p j2 · · · p j , where pi1 , pi2 , . . . , pik are distinct primes and p j1 , p j2 , . . . , p j are distinct primes. If u and v are adjacent, then all k +  primes are distinct and so {i 1 , i 2 , . . . , i k } ∩ { j1 , j2 , . . . , j } = ∅; while if u and v are not adjacent, then at least one of the primes pi1 , pi2 , . . . , pik is the same as one of the primes p j1 , p j2 , . . . , p j , that is, {i 1 , i 2 , . . . , i k } ∩ { j1 , j2 , . . . , j } = ∅. It therefore follows that the multi-prime labeling f can be considered as a function f : V (G) → P ∗ ([r ]) for some integer r ≥ 2, where P ∗ ([r ]) is the set of nonempty subsets of [r ], such that f (u) ∩ f (v) = ∅ if and only if u and v are adjacent vertices of G. Consequently, rather than assigning the integer f (u) = pi1 pi2 · · · pik to a vertex u of G, we can assign the subset f (u) = {i 1 , i 2 , . . . , i k } of [r ] to u. Furthermore, the minimum positive integer r for which such a function exists is the number ρ(G). When expressed in this manner, we refer to such a vertex labeling of a graph G as a subset labeling of G. Therefore, the concepts of multi-prime labeling and subset labeling are essentially the same concept and ρ(G) is the same parameter in each case. For example, three (essentially equivalent) multi-prime labelings and the corresponding subset labeling of a graph G of order 5 are shown in Figure 4.8. In the drawing of a graph, we write {a, b} as ab for simplicity. The first labeling uses the specific primes 2, 3, 5, the next two labelings refer to these three primes as p1 , p2 , p3 , and the fourth one describes the subscripts of these three primes as subsets of [3] (where, as indicated above, we write {1, 2} as 12). While the second labeling assigns the label p1 p2 to two vertices of G, the third labeling assigns distinct labels to distinct vertices of G. Since G contains K 3 as an induced subgraph, it follows by Observation 4.7 that ρ(G) = 3. As another example, consider the Petersen graph P of order 10. Figure 4.9 shows a subset labeling of P using labels in P ∗ ([5]). Thus, ρ(P) ≤ 5. Since P contains the 5-cycle as an induced subgraph, it follows that ρ(P) ≥ ρ(C5 ) = 5 by

Fig. 4.8 A graph G with ρ(G) = 3

4.3 Subset Labelings

39

Fig. 4.9 The Petersen graph P has ρ(P) = 5

Observation 4.7. Hence, ρ(P) = 5. An interesting feature of the subset labeling of P shown in Figure 4.9 is that this labeling uses all ten distinct 2-element subsets of [5] for the ten vertices of P. Actually, this vertex labeling of the Petersen graph is a familiar one. It is well known that   if we define a graph G = (V, E) such that the vertex set V of G is the set of all 25 = 10 distinct 2-element subsets of [5] and the edge set E is the set of all pairs of disjoint 2-element subsets of V , then the resulting graph G is isomorphic to the Petersen graph. The Petersen graph is a member of a special class of graphs. For positive  integers k and n with n > 2k, the Kneser graph K G n,k is that graph of order nk whose vertices are labeled with distinct elements of the set Pk ([n]) of k-element subsets of the set [n] where two vertices areadjacent if and only if they have disjoint labels. -regular. In particular, K G n,1 is the complete The graph K G n,k is therefore n−k k graph K n and K G 5,2 is the Petersen graph. The Kneser graphs are named for Martin Kneser, who, while investigating partitions of sets, made the following conjecture in 1955 [27] (stated in terms of graphs). Kneser’s Conjecture For positive integers k and n with n > 2k, there exists no proper (n − 2k + 1)-coloring of the Kneser graph K G n,k . Lovász [31] verified this conjecture in 1978 when he proved the following result. Theorem 4.9 For every two positive integers k and n with n > 2k, χ (K G n,k ) = n − 2k + 2. For integers n ≥ 2, the Kneser graphs On = K G 2n−1,n−1 , called odd graphs, have been of special interest. The graph O2 is K 3 , the graph O3 is the Petersen graph, and O4 is a 4-regular graph of order 35. By Theorem 4.9, every odd graph has chromatic number 3. We now investigate the multi-prime index ρ(G) for some specific graphs G. In each case, the labelings involved are subset labelings. We begin with paths. First, we evaluate ρ(G) when G is a small path. Proposition 4.3 For 3 ≤ n ≤ 6, ρ(Pn ) = n − 1.

40

4 Prime Labelings

Fig. 4.10 Subset labelings of P4 , P5 , and P6

Proof Clearly, ρ(P3 ) = 2. The subset labelings of Pn in Figure 4.10 show that ρ(Pn ) ≤ n − 1 for 4 ≤ n ≤ 6. Again, in the figure, the sets {a}, {a, b}, {a, b, c} are denoted by a, b, abc, respectively. We only show that ρ(P6 ) = 5. If f is a subset labeling of P6 = (v1 , v2 , . . . , v6 ) using subsets in [4], then there is no vertex v of P6 for which | f (v)| = 1, for suppose that f (v j ) = {1} where 1 ≤ j ≤ 6. Then 1 ∈ f (u) if u is not a neighbor of v j . Since at least two such vertices are adjacent, this is impossible. There is no vertex v of P6 with | f (v)| = 3 either, for this would imply that the label of each neighbor of v is a singleton. Consequently, the label of each vertex of P6 is a 2-element subset of [4]. However, this implies that f (vi ) = {1, 2}, say, where i is odd and f (vi ) = {3, 4}, where i is even. Since v1 and v4 are not   adjacent and f (v1 ) ∩ f (v4 ) = ∅, this is a contradiction. Hence, ρ(P6 ) = 5. In fact, 6 is the largest value of n for which ρ(Pn ) = n − 1. In order to show this, we first present the following result. Theorem 4.10 For an integer n ≥ 7, let Pn = (v1 , v2 , . . . , vn ) and let k be an integer with k ≥ ρ(Pn ). Suppose that there is a subset labeling f n : V (Pn ) → P ∗ ([k]) for which there are two distinct integers a and b such that a ∈ f n (vi ) for each even integer i and b ∈ f n (vi ) for each odd integer i where 1 ≤ i ≤ n. Then there is a subset labeling f n+1 : V (Pn+1 ) → P ∗ ([k + 1]) of Pn+1 = (v1 , v2 , . . . , vn+1 ) such that a ∈ f n+1 (vi ) for each even integer i and b ∈ f n+1 (vi ) for each odd integer i where 1 ≤ i ≤ n + 1. Proof For Pn+1 = (v1 , v2 , . . . , vn , vn+1 ), define the labeling f n+1 : V (Pn+1 ) → P ∗ ([k + 1]) from the labeling f n of Pn as follows: 

If n is odd, then ⎧ if i is even for 2 ≤ i ≤ n − 1 or i = n ⎨ f n (vi ) f n+1 (vi ) = f n (vi ) ∪ {k + 1} if i is odd for1 ≤ i ≤ n − 2 ⎩ {a, k + 1} if i = n + 1.



If n is even, then ⎧ if i is odd for 1 ≤ i ≤ n − 2ori = n ⎨ f n (vi ) f n+1 (vi ) = f n (vi ) ∪ {k + 1} if i is even for 2 ≤ i ≤ n − 2 ⎩ {b, k + 1} if i = n + 1.

4.3 Subset Labelings

41

Fig. 4.11 Subset labelings of Pn for n = 7, 8, 9

Fig. 4.12 A subset labeling of P11

Then f n+1 is a subset labeling of Pn+1 with the desired properties.

 

Figure 4.11 shows a subset labeling of P7 , which implies that ρ(P7 ) ≤ 5. However, since ρ(P6 ) = 5, it follows that ρ(P7 ) = 5. For n = 7 and n = 8, the subset labelings f n+1 of P8 and P9 in Figure 4.11 are those defined in the proof of Theorem 4.10. As before, we write {a, b, c, . . .} as abc · · · . The following is a consequence of Theorem 4.10. Corollary 4.2 For each integer n ≥ 7, ρ(Pn ) ≤ n − 2. Proof The subset labeling f 7 : V (P7 ) → P ∗ ([5]) of P7 shown in Figure 4.11 has the desired property that 1 ∈ f 7 (vi ) for each even integer i and 3 ∈ f n (vi ) for each odd integer i where 1 ≤ i ≤ 7. Repeatedly applying Theorem 4.10, we obtain, for an integer n ≥ 7, a subset labeling f n : V (Pn ) → P ∗ ([n − 2]) of Pn = (v1 , v2 , . . . , vn ) such that 1 ∈ f n (vi ) for each even integer i and 3 ∈ f n (vi ) for each odd integer i,   where 1 ≤ i ≤ n. Therefore, ρ(Pn ) ≤ n − 2 for n ≥ 7. By Corollary 4.2, ρ(P8 ) ≤ 6; in fact, ρ(P8 ) = 6. However, 8 is the largest value of n for which ρ(Pn ) = n − 2. Proposition 4.4 For 8 ≤ n ≤ 11, ρ(Pn ) = 6. Proof Figure 4.12 shows a subset labeling f : V (P11 ) → P ∗ ([6]) of P11 and so ρ(P11 ) ≤ 6. It then follows by Observation 4.7 that 6 = ρ(P8 ) ≤ ρ(Pn ) ≤ 6 for 8 ≤ n ≤ 11. Hence, it remains to show that ρ(P8 ) ≥ 6. Assume, to the contrary, that ρ(P8 ) < 6. Since ρ(P7 ) = 5, it follows that ρ(P8 ) = 5 and so there is a subset labeling g : V (P8 ) → P ∗ ([5]) of P8 . Observe that (1) no vertex of P8 can be labeled with a single integer,

42

4 Prime Labelings

(2) no vertex of P8 can be labeled with four or more integers, and (3) no interior vertex of P8 can be labeled with three integers. Let P8 = (v1 , v2 , . . . , v8 ). If |g(v1 )| = 3, say g(v1 ) = {1, 2, 3}, then we may assume that g(v2 ) = {4, 5}, g(v3 ) = {1, 2}, g(v4 ) = {3, 4}, g(v5 ) = {1, 5}, g(v6 ) = {2, 4}. However then, g(v7 ) = {1, 3, 5}, which contradicts (3). If |g(v1 )| = 2, say g(v1 ) = {1, 2}, then we may assume that g(v2 ) = {3, 4}, g(v3 ) = {1, 5}, g(v4 ) = {2, 3},   g(v5 ) = {1, 4}. However then, g(v6 ) = {2, 3, 5}, which again contradicts (3). Since Pn is an induced subgraph of Pn+1 for each positive integer n, it follows by Observation 4.7 that ρ(Pn ) ≤ ρ(Pn+1 ) and so {ρ(Pn )}, n ≥ 2, is a nondecreasing sequence of positive integers. First, we establish another property of this sequence. Theorem 4.11 For an integer n ≥ 3, ρ(Pn ) ≤ ρ(Pn+1 ) ≤ ρ(Pn ) + 1. Proof That ρ(Pn ) ≤ ρ(Pn+1 ) follows from Observation 4.7, as noted above. We now show that ρ(Pn+1 ) ≤ ρ(Pn ) + 1. Let Pn = (v1 , v2 , . . . , vn ) and Pn+1 = (v1 , v2 , . . . , vn , vn+1 ). Suppose that ρ(Pn ) = r . Let g : V (Pn ) → P ∗ ([r ]) be a multi-prime labeling of Pn . We now define a subset labeling f : V (Pn+1 ) → P ∗ ([r + 1]) by ⎧ if 1 ≤ i ≤ n with i = n − 2 ⎨ g(vi ) f (vi ) = g(vi ) ∪ {r + 1} if i = n − 2 ⎩ [r + 1] − g(vn ) if i = n + 1. Therefore, f (vi ) ∩ f (vi+1 ) = ∅ for 1 ≤ i ≤ n and f (vi ) ∩ f (v j ) = ∅ for 1 ≤ i, j ≤ n and |i − j| ≥ 2. It remains to show that f (vn+1 ) ∩ f (vi ) = ∅ for 1 ≤ i ≤ n − 1. Since f (vn ) ∩ f (vn−1 ) = ∅ and f (vn+1 ) = [r + 1] − g(vn ), there exists an integer j ∈ [r ] such that j ∈ f (vn+1 ) ∩ f (vn−1 ). Thus, f (vn+1 ) ∩ f (vn−1 ) = ∅. Since r + 1 ∈ f (vn+1 ) ∩ f (vn−2 ), it follows that f (vn+1 ) ∩ f (vn−2 ) = ∅. For each integer i with 1 ≤ i ≤ n − 3, there exists an integer i ∈ f (vi ) such that i ∈ f (vn−1 ). Since f (vn−1 ) ∩ f (vn ) = ∅, it follows that i ∈ f (vn+1 ). Thus, f (vn+1 ) ∩ f (vi ) = ∅ for 1 ≤ i ≤ n − 3. Hence, f is a multi-prime labeling of Pn+1 and so ρ(Pn+1 ) ≤ r + 1 =   ρ(Pn ) + 1. The following result establishes another property of the sequence {ρ(Pn )}. Theorem 4.12 lim ρ(Pn ) = ∞. n→∞

Proof Assume, to the contrary, that limn→∞ ρ(Pn ) = ∞. By Theorem 4.11, there exist positive integers N and r such that ρ(Pn ) = r for every n ≥ N . Hence, there is a sufficiently large positive integer n such that for the path Pn = (v1 , v2 , . . . , vn ) and a multi-prime labeling f : V (Pn ) → P ∗ ([r ]), there are distinct vertices vi , v j , vk of Pn with i < j < k for which f (vi ) = f (v j ) = f (vk ) = S ⊆ [r ]. Thus, j − i ≥ 2 and k − j ≥ 2. Since f (vi ) ∩ f (vk−1 ) = ∅, there exists  ∈ f (vi ) ∩ f (vk−1 ). Thus,  ∈ S. However then,  ∈ f (vk−1 ) ∩ f (vk ) and so f (vk−1 ) ∩ f (vk ) = ∅, a contradiction.  

4.3 Subset Labelings

43

Fig. 4.13 Subset labelings of C6 and C7

The following is a consequence of Theorems 4.11 and 4.12. Corollary 4.3 For each integer r ≥ 3, there exists an integer n r such that ρ(Pnr ) = r. We now turn our attention to cycles. We have seen that ρ(C3 ) = 3, ρ(C4 ) = 2, and ρ(C5 ) = 5. Proposition 4.5 ρ(C6 ) = 5 and ρ(C7 ) = 7. Proof The subset labelings of C6 and C7 in Figure 4.13 show that ρ(C6 ) ≤ 5 and ρ(C7 ) ≤ 7. We only show that ρ(C7 ) = 7. Let C7 = (v1 , v2 , . . . , v7 , v1 ). Assume, to the contrary, that there is a subset labeling f : V (C7 ) → P ∗ ([6]) of C7 . We begin with some observations. (1) There is no vertex v of C7 with | f (v)| = 1. Suppose that f (v1 ) = {1}. Since v3 and v4 are not adjacent to v1 , it follows that 1 ∈ f (v3 ) and 1 ∈ f (v4 ). However, v3 and v4 are adjacent and f (v3 ) ∩ f (v4 ) = ∅, a contradiction. (2) There is no vertex v of C7 with | f (v)| = 5. Suppose that f (v1 ) = {1, 2, 3, 4, 5}. Thus, f (v2 ) = 6, which is impossible by (1). (3) There is no vertex v of C7 with | f (v)| = 4. Suppose that f (v1 ) = {1, 2, 3, 4}. Since f (v1 ) ∩ f (v2 ) = ∅ and f (v1 ) ∩ f (v7 ) = ∅, it follows by (1) that f (v2 ) = f (v7 ) = {5, 6}. Since f (v2 ) ∩ f (v3 ) = ∅, it follows that f (v3 ) ⊆ {1, 2, 3, 4} and so f (v3 ) ∩ f (v7 ) = ∅. However, v3 and v7 are not adjacent. This is a contradiction. (4) There do not exist adjacent vertices u and v of C7 with | f (u)| = | f (v)| = 3. Suppose that f (v1 ) = {1, 2, 3} and f (v2 ) = {4, 5, 6}. Since f (v2 ) ∩ f (v3 ) = ∅, it follows that f (v3 ) ⊆ {1, 2, 3} and f (v7 ) ⊆ {4, 5, 6} and so f (v3 ) ∩ f (v7 ) = ∅, again a contradiction. (5) There do not exist vertices u and v of C7 having a common neighbor such that | f (u)| = | f (v)| = 2. Suppose that | f (v1 )| = | f (v3 )| = 2 and f (v1 ) = {1, 2}. We may assume that 1 ∈ f (v3 ), 2 ∈ f (v4 ), 1 ∈ f (v5 ), and 2 ∈ f (v6 ). Since | f (v3 )| = 2 and 2 ∈ f (v4 ), we may further assume that f (v3 ) = {1, 3}. Because

44

4 Prime Labelings

v3 is adjacent to neither v6 nor v7 and 1 ∈ f (v5 ) ∩ f (v1 ), it follows that 3 ∈ f (v6 ) and 3 ∈ f (v7 ). However, v6 and v7 are adjacent, which is impossible. By (1)–(3), | f (v)| = 2 or | f (v)| = 3 for every vertex v of C7 . However, (4) and (5) imply that no such labeling exists.   For each integer n ≥ 3, the path Pn−2 is an induced subgraph of the n-cycle Cn . Thus, ρ(Pn−2 ) ≤ ρ(Cn ) by Observation 4.7. Therefore, the following is a consequence of Theorem 4.12. Proposition 4.6 lim ρ(Cn ) = ∞. n→∞

While lim ρ(Cn ) = ∞ by Proposition 4.6, it follows by Proposition 4.5 that n→∞ there is no result for cycles that corresponds to Theorem 4.11 for paths. The results mentioned above suggest the following questions. Problem 4.2 What is ρ(Pn ) for each integer n ≥ 2? Problem 4.3 What is ρ(Cn ) for each integer n ≥ 3? We have now seen examples of graphs whose multi-prime index equals its chromatic number and others whose multi-prime index exceeds its chromatic number. As we now show, there is no other possibility. Theorem 4.13 If G is a nontrivial connected graph, then ρ(G) ≥ χ (G). Proof Let ρ(G) = r ≥ 2 and let f : V (G) → P ∗ ([r ]) be a multi-prime labeling of G. Define the vertex coloring c : V (G) → [r ] by c(x) = min{i ∈ [r ] : i ∈ f (x)}. Let u and v be two adjacent vertices of G. Since f (u) ∩ f (v) = ∅, it follows that c(u) = c(v). Thus, c is a proper coloring of G using at most r colors. Therefore, χ (G) ≤ r = ρ(G).   We have seen that ρ(G) ≥ 2 for every nonempty connected graph G. Next, we present a characterization of connected graphs G having ρ(G) = 2. Proposition 4.7 Let G be a nontrivial connected graph. Then ρ(G) = 2 if and only if G is a complete bipartite graph. Proof Since ρ(G) = 2 for every complete bipartite graph G by Corollary 4.1, it remains to verify the converse. Let G be a nontrivial connected graph with ρ(G) = 2. By Theorem 4.13, G is bipartite and no vertex of G can be labeled {1, 2}. Thus, every vertex of G is labeled {1} or {2}. Let U be the set of vertices of G labeled {1} and W be the set of vertices of G labeled {2}. Therefore, U and W are independent sets. Since {1} and {2} are disjoint, every vertex of U is adjacent to every vertex of W . That is, G is a complete bipartite graph with partite sets U and W .  

4.3 Subset Labelings

45

Fig. 4.14 Two graphs with multi-prime index 3

According to Proposition 4.7, if G is a nontrivial connected graph that is not a complete bipartite graph, then ρ(G) ≥ 3. We saw that ρ(G) = 3 for the graph G of Figure 4.8 and that the multi-prime index of every complete 3-partite graph is 3. There are other graphs having multi-prime index 3. We saw in Proposition 4.3 that ρ(P4 ) = 3; also, ρ(K 3 ) = 3. Furthermore, the graph G of Figure 4.14 also has multiprime index 3. Multi-prime labelings of P4 and G are shown in the figure. By Proposition 4.2, a composition graph can be constructed from P4 and G by replacing each vertex vi by an empty graph, resulting in a graph having the same multi-prime index as P4 or K 3 . It follows from Theorem 4.13 that if G is a connected graph with ρ(G) = 3, then χ (G) = 2 or χ (G) = 3. Therefore, there are graphs G with ρ(G) = 3 for which χ (G) = k where k = 2 or k = 3. Similarly, if G is a connected graph with ρ(G) = 4, then for each integer k ∈ {2, 3, 4}, there exists a connected graph G with ρ(G) = 4 and χ (G) = k. This gives rise to the following conjecture. Conjecture 4.2 For each pair k,  of integers with 2 ≤ k ≤ , there exists a connected graph G with χ (G) = k and ρ(G) = .

4.4 2-Prime Labelings We saw in a multi-prime labeling of a graph G that integers from the set [2, ∞) are assigned to the vertices of G in such a way that two vertices of G are assigned relatively prime integers if and only if they are adjacent. The complement G of G therefore has the property that two distinct integers a and b from the set [2, ∞) can be assigned to adjacent vertices of G if and only if a and b are not relatively prime, that is, if there is some prime p such that p | a and p | b. Both the 5-cycle C5 and the wheel W5 = C4 ∨ K 1 of Figure 4.15 have such a labeling. In both labelings, each vertex label is the product of two distinct primes. These are examples of a particular kind of labeling having a property that is opposite to that of a multi-prime labeling. An integer is a 2-prime integer if it is the product of two distinct primes. A 2-prime labeling of a graph G is an assignment of distinct 2-prime integers to the vertices of G such that two vertices of G are assigned labels that are not relatively prime if and only if they are adjacent. Thus, in a 2-prime labeling f of a graph, every two distinct

46

4 Prime Labelings

Fig. 4.15 Labeling the vertices of C5 and W5 = C4 ∨ K 1 Fig. 4.16 A graph with no 2-prime labeling

vertices u and v are assigned two distinct 2-prime integers. If u and v are adjacent, then f (u) and f (v) are not relatively prime; while if u and v are not adjacent, then f (u) and f (v) are relatively prime. Therefore, both labelings shown in Figure 4.15 are 2-prime labelings. Not all graphs can be given 2-prime labelings, however, for consider the graph G of Figure 4.16. Suppose that there is a 2-prime labeling f of G. Then f (z) = p1 p2 , where p1 and p2 are distinct primes. Since vz ∈ E(G), we can assume that f (v) = p1 p3 , where p3 is a prime distinct from p1 and p2 . Since yz ∈ E(G), either / E(G), it follows p1 | f (y) or p2 | f (y), but not both. Since f (v) = p1 p3 and vy ∈ that p1  f (y); thus, p2 | f (y). Hence, f (y) = p2 p4 for some prime p4 distinct from p1 and p3 . Because wz ∈ E(G), either p1 | f (w) or p2 | f (w), both of which are impossible since wv, wy ∈ / E(G). Therefore, no 2-prime labeling of G is possible. The argument we gave as to why there is no 2-prime labeling of the graph G of Figure 4.16 comes from the fact that G contains G[{z, v, w, y}] = K 1,3 as an induced subgraph and there is no 2-prime labeling of K 1,3 . In a similar manner, one can show that there is no 2-prime labeling of any of the nine graphs shown in Figure 4.17. Thus, no graph containing any of these nine graphs as induced subgraphs has a 2-prime labeling. These nine graphs play a key role in another concept in graph theory. Let G be a nonempty graph. The line graph L(G) of G is that graph whose vertex set consists of the edges of G, where two vertices of L(G) are adjacent if the corresponding edges of G are adjacent. A graph H and its line graph L(H ) are shown in Figure 4.18. A graph G is a line graph if there exists a graph F such that G = L(F). Thus, the graph L(H ) of Figure 4.18 is a line graph. In 1970, Lowell Beineke [2] characterized the graphs that are line graphs as those graphs containing none of the nine graphs shown in Figure 4.17 as induced subgraphs. Theorem 4.14 (Beineke’s Theorem) A graph G is a line graph if and only if G does not contain any of the nine graphs of Figure 4.17 as an induced subgraph.

4.4 2-Prime Labelings

47

Fig. 4.17 The nine forbidden induced subgraphs of a line graph

In particular, the graph G of Figure 4.16 is not a line graph. Theorem 4.14 allows us to present the following result. Theorem 4.15 A graph G possesses a 2-prime labeling if and only if G is a line graph. Proof Assume first that G is a graph that possesses a 2-prime labeling. Then each induced subgraph of G has a 2-prime labeling as well. Since none of the nine graphs of Figure 4.17 has a 2-prime labeling, it follows that G contains none of these nine graphs as an induced subgraph. By Theorem 4.14, G is a line graph. Conversely, assume that G is a line graph. Then there is a graph H such that G = L(H ). Suppose that H has order n. Label the n vertices of H with n distinct primes p1 , p2 , . . . , pn . For an edge e of H joining vertices labeled pi and p j , where 1 ≤ i, j ≤ n and i = j, we assign the 2-prime integer pi p j to the edge e. Then the vertex e of G = L(H ) is also labeled pi p j . This produces a vertex labeling of G. Observe that two vertices e and e of G are adjacent in G if and only if the edges e and e of H are incident with a common vertex, labeled pk say, where then pk divides the labels of e and e . Therefore, this vertex labeling is a 2-prime labeling of G.   As is the case for multi-prime labelings, 2-prime labelings can also be expressed as a certain type of subset labeling. For example, if G is a graph with a 2-prime labeling in which the primes involved are p1 , p2 , . . . , pn , then each vertex labeled pi p j can be expressed as the 2-element subset {i, j} of [n]. For example, for the 2-prime labelings of the graphs C5 and W5 = C4 ∨ K 1 shown in Figure 4.15, we can write p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11. Expressing the 2-prime integer pi p j as the 2-element subset {i, j} of [4] or [5], which we write again as i j, we have

48

4 Prime Labelings

Fig. 4.18 A graph and its line graph Fig. 4.19 2-subset labelings of C5 and W5 = C4 ∨ K 1

the labelings of these two graphs shown in Figure 4.19. A 2-element subset of [n] for some integer n ≥ 2 is referred to as a 2-subset. A 2-subset labeling of a graph G is an assignment of distinct 2-subsets to the vertices of G such that the labels of two vertices have a nonempty intersection if and only if these vertices are adjacent. Thus, the labelings of C5 and W5 in Figure 4.19 are 2-subset labelings. Consequently, the concept of 2-subset labelings of graphs provides us with an alternative way of looking at 2-prime labelings, giving the following corollary of Theorem 4.15. Corollary 4.4 A graph G possesses a 2-subset labeling if and only if G is a line graph. Therefore, not only are the graphs C5 and W5 line graphs (in fact, C5 = L(C5 ) and W5 = L(K 1,1,2 )) but the complement P of the Petersen graph P is a line graph. In fact, P = L(K 5 ). It is useful to describe another concept in graph theory. For a nonempty set A and a collection S of distinct nonempty subsets of A, the intersection graph Ω(S) of S is that graph with vertex set S where two vertices are adjacent if and only if the subsets have a nonempty intersection. For example, let G be a nontrivial connected graph with V (G) = {v1 , v2 , . . . , vn }, where A = E(G), Si is the set of edges incident with vi for 1 ≤ i ≤ n, and S = {S1 , S2 , . . . , Sn }. Then Ω(S) ∼ = G, that is, every nontrivial connected graph is an intersection graph. Thus, each line graph is an intersection graph where the subsets involved are 2-subsets. The topic of intersection graphs has been discussed in detail in a book by Terry A. McKee and F. R. McMorris [32]. Integers that are k-prime integers and k-prime labelings of graphs for an integer k ≥ 2 are defined as expected. An integer is a k-prime integer, k ≥ 2, if it is the product of k distinct primes. A k-prime labeling of a graph G is an assignment of distinct k-prime integers to the vertices of G such that the labels of two vertices u

4.4 2-Prime Labelings

49

Fig. 4.20 A 3-prime labeling of the Petersen graph

and v of G are not relatively prime if and only if u and v are adjacent. For example, a 3-prime labeling of the Petersen graph is shown in Figure 4.20 (where the 15 distinct primes here are denoted by a, b, c, d, e, f, g, h, i, j, k, l, p, q, r). Since the Petersen graph contains K 1,3 as an induced subgraph, this graph has no 2-prime labeling (and so the Petersen graph is not a line graph). As expected, the concepts of k-prime integers and k-prime labelings can be looked at in terms of subsets. For integers k and n with 2 ≤ k ≤ n, a k-element subset of [n] is referred to as a k-subset. A k-subset labeling of a graph G is an assignment of distinct k-subsets to the vertices of G such that the labels of two vertices u and v have a nonempty intersection if and only if u and v are adjacent. The following observation is immediate. Observation 4.16 For an integer k ≥ 2, a graph G possesses a k-prime labeling if and only if G has a k-subset labeling. As Figure 4.20 shows, the Petersen graph has a 3-subset labeling. While none of the nine graphs of Figure 4.17 has a 2-subset labeling, each has a 3-subset labeling, examples of which are shown in Figure 4.21 (where the set [12] of the first 12 positive integers is denoted by {1, 2, . . . , 9, x, y, z} and a 3-subset {a, b, c} is once again denoted by abc in Figure 4.21). In fact, every nontrivial connected graph has a k-subset labeling for some integer k ≥ 2, as we show next. Theorem 4.17 Every connected graph of order n ≥ 3 has a k-subset labeling for some integer k with 2 ≤ k < n. Proof We proceed by induction on the order n of a connected graph. It is straightforward to show that the statement is true for all connected graphs of order 3 or 4. Assume that every connected graph of order n, where n ≥ 4, has a k-subset labeling for some integer k where 2 ≤ k < n. Let G be a connected graph of order n + 1 and let v be a vertex of G that is not a cut-vertex of G, where degG v = d. Thus, G − v is a connected graph of order n. By the induction hypothesis, G − v has a k-subset labeling for some integer k with 2 ≤ k < n. We may assume that all vertex labels are k-subsets of [r ] for some positive integer r . Let V (G) − {v} = {v1 , v2 , . . . , vn },

50

4 Prime Labelings

Fig. 4.21 3-subset labelings of the nine graphs in Figure 4.17

where NG (v) = {v1 , v2 , . . . , vd }, and let Si , 1 ≤ i ≤ n, be the k-subset of [r ] assigned to vi . Now, define Ti = Si ∪ {r + i} for 1 ≤ i ≤ n and T = {r + i : 1 ≤ i ≤ d}. Thus, |Ti | = k + 1 ≤ n for 1 ≤ i ≤ n and |T | = d ≤ n.  If d = k + 1, then we assign Ti to vi for 1 ≤ i ≤ n and T to v, producing a (k + 1)-subset labeling of G.  If d < k + 1, then let T  = T ∪ {r + n + i : 1 ≤ i ≤ k + 1 − d}. Thus, |T  | = d + (k + 1) − d = k + 1. We assign Ti to vi for 1 ≤ i ≤ n and T  to v, producing a (k + 1)-subset labeling of G.  If d > k + 1, then let A = {r + n + i : 1 ≤ i ≤ n(d − k − 1)}. Thus, |A| = n(d − k − 1). Let {A1 , A2 , . . . , An } be a partition of A where |Ai | = d − k − 1 for 1 ≤ i ≤ n. Define Ti = Ti ∪ Ai for 1 ≤ i ≤ n. Then |Ti | = (k + 1) + (d − k − 1) = d = |T | for 1 ≤ i ≤ n. We assign Ti to vi for 1 ≤ i ≤ n and T to v, producing a d-subset labeling of G.   Theorem 4.17 suggests the following concept. For a connected graph G of order n ≥ 3, let ψ(G) denote the minimum positive integer k for which G has a k-subset labeling. Thus, if ψ(G) = k, then G has a j-subset labeling for every integer j ≥ k but no j-subset labeling for any integer j with j < k. Furthermore, if G is a graph with ψ(G) = k ≥ 3 and v is a non-cut-vertex of G, then ψ(G − v) ≥ k − 1.

4.4 2-Prime Labelings

51

Fig. 4.22 Two graphs G with ψ(G) = 4 but ψ(G − v) = 3 for every non-cut-vertex v of G

Necessarily, if G is a connected graph of order 3 or more with ψ(G) = 2 and v is a non-cut-vertex of G, then ψ(G − v) = 2 as well. The graphs in Figure 4.21 all have the property that ψ(G) = 3 and ψ(G − v) = 2 for every non-cut-vertex v of G. The fact that none of the nine graphs of Figure 4.17 has a 2-subset labeling but each has a 3-subset labeling suggests the following concept. For an integer k ≥ 2, a graph G is a k-subset graph if G possesses a k-subset labeling. Thus, a graph G is a 2-subset graph if and only if G is a line graph. So, k-subset graphs with k ≥ 3 may be considered generalizations of line graphs. While the nine graphs of Figure 4.17 constitute a forbidden subgraph characterization of 2-subset graphs, this suggests the problem of determining the class S of forbidden subgraphs for 3-subset graphs. Each such graph G in this class S then has the property that ψ(G) = 4 but ψ(G − v) = 3 for every non-cut-vertex v of G. Two members of S are shown in Figure 4.22. It is useful to make one other comment in this chapter. Suppose that G is a k-subset graph for some integer k ≥ 2. Then G possesses a k-subset labeling f . Therefore, f assigns distinct k-subsets to the vertices of G where two vertices are assigned nondisjoint k-subsets if and only if they are adjacent. Consequently, for the complement G of the graph G, the labeling f assigns disjoint k-subsets to two vertices of G if and only if these vertices are adjacent in G. Graphs with this property have been studied by many where these labelings were referred to as k-tuple colorings. This coloring concept was introduced by Saul Stahl [44] in 1976.

Chapter 5

Additive Labelings

In the preceding chapters, our interest has been on various types of vertex labelings of graphs, where typically the vertex labels are nonnegative integers. In this chapter, our emphasis initially shifts to edge labelings, where each edge labeling generates a vertex coloring possessing some desired property. Later in the chapter, we move on to vertex labelings that produce vertex colorings. All of these labelings have something in common—namely, the resulting vertex coloring is obtained by summing certain labels. In addition, rather than having distinct labels, as has often been the case, the goal here is to produce a desired coloring by permitting duplication of labels and minimizing the largest integer used as a label.

5.1 Rainbow Additive Labelings It is well known (in fact part of graph theory folklore) that every nontrivial graph has at least two vertices having the same degree. Indeed, for every integer n ≥ 2, there is only one connected graph G n having exactly two vertices of the same degree. The graphs G 2 , G 3 , and G 4 are shown in Figure 5.1, where the vertices are labeled with their degrees. Of course, the complement G n of G n also has exactly two vertices of the same degree, although G n is disconnected. A graph G has been called irregular if no two vertices of G have the same degree. As we just stated, no nontrivial graph is irregular. A graph having exactly two vertices of the same degree is called nearly irregular. So, all three graphs of Figure 5.1 are nearly irregular. The situation is quite different with multigraphs, however. For example, each of the three multigraphs in Figure 5.2 is irregular. In each case, the vertices are labeled with their degrees. For i = 3, 4, the irregular multigraph Mi is obtained from the graph G i of Figure 5.1 by the addition of a single edge. The three multigraphs in Figure 5.2 can be represented in another manner. Figure 5.3 shows three graphs G 1 , G 3 , and G 4 (where, once again, G 3 and G 4 are the graphs in Figure 5.1). The edges of these three graphs are labeled with elements of the set [3]. In each case, the label assigned to an edge uv corresponds to the number © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6_5

53

54

5 Additive Labelings 2

1...

1....

1 ....

............

G2 :

G3 : . ..............

1

........... ............ ... .. ... ... ... .. . . ... .. . ... ... ... ... ... ... .... ... .. . .......... ........

G4 :

2

. ............... ............... ... ... ... ... ... . . . ... ... ... ... .... ............ .......

3

.... ...........

2

1

Fig. 5.1 Connected graphs having exactly two vertices of the same degree 2

3

3

........ ................. ...... .. .. ..... ... ..... ... .... ... . ... ... ... ... ... ... ... ... .... ..... ... .. . ......... ............................................. .... .. ............ ........ ....... . . . . .............................. ..

M1 :

5

1 ......

2 .......

......... .......... ... ... ... .... ..... ... ... ... .. . ... ... .. .. ... ... ... ... ... ... ... ... ..... ... ... ..... .. ... ............... ........

M3 :

M4 :

. ............... .............. .. . ... ..... ..... ... ... .... . . .. ... ... ... .... ... ...... ............ ............ .. .....

4

. ..............

1

3

4

Fig. 5.2 Irregular multigraphs

of parallel edges joining u and v in the associated multigraph of Figure 5.2. From each such edge labeling, a vertex labeling is obtained by summing the labels of the edges incident with the vertex. Each vertex label therefore results in the degree of the vertex in the associated multigraph. This corresponds to an edge labeling which will be a major topic of discussion in this chapter. An edge labeling f : E(G) → [k] of a graph G that results in the vertex labeling f  : V (G) → N defined by 

f  (v) =

f (uv)

u∈N (v)

is referred to as an additive labeling of G. If no two vertices of G have the same label, then f is a rainbow additive labeling of G. Thus, the labelings of the graphs in Figure 5.3 are rainbow additive labelings. The term rainbow is often used in graph theory to denote colorings where the vertices (or edges) in a graph or subgraph have different colors. That is, a rainbow additive labeling of a graph is an edge labeling

...... ..... ....... ..... .. ..................... . ... ... ... .. . ... ... ... ... ... ... .. . ..... . . . . . . ........... .. ..... .... ..... . ..... . ... . ... ..... ..... ....... ............ ......

3

G1 :

2

1

4

5

3

Fig. 5.3 The graphs G 1 , G 2 , G 3

G3 :

......... ........ .... ...... .... ....... .... .... . ..... ...... ..... ...... ........ ........ ... ... ... .. .. ... . . . ... ... ... ... .............. .... ..... .. ..... ..... ...... .......

2

1

2

1

3

....... ........ .... ....... ..... ....... .... .... . . ..... ...... ..... ...... ........... ....... . . . . ..... . ..... ......... ......... ......... ......... ... ... . ... ................

1

3

2

2

G4 :

1

4

1

...... ..... ....... . ..... ..... ...... .......

1

5.1 Rainbow Additive Labelings

55

that produces a rainbow vertex coloring. For this reason, we refer to a vertex coloring here rather than a vertex labeling. A rainbow additive labeling of a graph G is similar to a type of labeling called an antimagic labeling. In the case of an antimagic labeling, however, if G has size m, then the edges are labeled with distinct elements of the set [m]. If G is a connected graph of order 3 or more, then G always has a rainbow additive labeling. For example, if E(G) = {e1 , e2 , . . . , em }, then the function f : E(G) → [2m−1 ] defined by f (ei ) = 2i−1 for 1 ≤ i ≤ m is one such rainbow additive labeling of G (since no two vertices have the same set of incident edges). The minimum positive integer k for which G has a rainbow additive labeling f : E(G) → [k] is the rainbow additive index ra(G) of G. This concept was introduced by Gary Chartrand in 1986 [6] at the 250th Anniversary of Graph Theory Conference held at Indiana University-Purdue University Fort Wayne (now called Purdue University Fort Wayne). At that time and in succeeding years, this concept has been studied and referred to using different terminologies. In fact, the rainbow additive index ra(G) of a graph G has often been called the irregularity strength of G, as the strength of a multigraph M is the maximum number of parallel edges joining any two vertices of M. Since there is no irregular graph, ra(G) ≥ 2 for every connected graph of order 3 or more. In fact, there is an infinite class of graphs G, already introduced, for which ra(G) = 2. Proposition 5.1 For every nearly irregular graph G n of order n ≥ 3, ra(G n ) = 2. Proof The rainbow additive labelings of the graphs G 3 and G 4 in Figure 5.3 show that ra(G 3 ) = ra(G 4 ) = 2. Interchanging the labels 1 and 2 also results in a rainbow additive labeling of G 4 (see Figure 5.4). The labeling of G 5 in Figure 5.4 is also a rainbow additive labeling, which shows that ra(G 5 ) = 2. For a nearly irregular graph G n of order n ≥ 3 with V (G n ) = {v1 , v2 , . . . , vn } such that deg v1 ≤ deg v2 ≤ . . . ≤ deg vn , the degrees of these vertices are as follows:  degG n vi =

v2 3

5 v 2

3

2

G5 :

... ...... ......... . ..... ..... ....... ......

2

v1

Fig. 5.4 Rainbow additive labelings of G 4 and G 5

v2

2

......... ........ .... ....... .... ...... .... ... . ..... ...... .................... . . . ....... ...... . . ...... ..... ...... ..... . . . . ...... .. ...... ..... ...... ...................... ... ...... .. .... . . ....... ...... 5 ...... .......... ........... ...... ...... ...... ..... ...... ...... . . . . . .................. .................... . . ... .... .. .... ..... .. ..... ...... ..... ...... ...... ......

6

4

2

1

G4 :

v4

v3

........ ...... .... ...... ..... ....... . .... ..... . ..... ....... ..... ....... ....... .......... ...... ...... ...... ...... . . . . . ...... .. ...... .......................... ...... .... ... . ..... ...... ...... 4

2

  i if 1 ≤i ≤ n2 i − 1 if n2 + 1 ≤ i ≤ n.

2

1

7

2

4 v3

v

2

2

v1

56

5 Additive Labelings 3

..... .......... ..... ....... .... ...... .. .... ..... ...... ...... ..... .......... . . . . ....... ..... .. ....... ... ....... ...... ..... ...... .............. ........... ... ........ ... ........ ... ... . ... ... ... ... ..... ....... ... ..... . ... . ... ... ... ... ... ... ... ... ... ....... ..... ........ ........ .. .... ..... ...... .......

8

3

5

K4 :

1

6

2

1

1

3

K5 :

9

2

1

7

1

4

1

....... ....... ..... ....... ..... ....... ... .... . . ..... .............. ...................... . . . . . . ............. ........ ....... ........ .... ........ ... ..... ........ ............. .............. . ... . ... ...... . . . . . . . ........ .......... ... ..... .... .. .... ... ..... . .... ... .. ... ..... ... ..... .. .. .............. ... ..... .. .. ... ......... .... ..... . . . . . ..... .. ... .. ... ..... ... ... ..... ...... ..... ... ... .... ............ ...... ... ... .. ... .... ........ ......... . . . . ... .. ... ... ... ... ...... ........ ... .. ... ... ... ... ... ... ... .. . . ... ... . ... ............... .... ... .. ..... .... ... .... ....... ........

1

1

3

3

6

3

3

12

Fig. 5.5 Rainbow additive labelings of K 4 and K 5

In particular, the vertices v n2  and v n2 +1 are the only two vertices of G n having the same degree. The vertex vn is adjacent to all other vertices of G n . The labeling that assigns the label 2 to all edges except the edge vn v n2  , which is labeled 1, results     in f  (vi ) = 2 degG n vi for i = n2 , n and f  (vi ) = 2 degG n vi − 1 for i = n2 , n. n Since the vertices vi for i = 2 , n have distinct even degrees in G n , they have distinct even colors. Since f  (vn ) = f  (v n2  ) and both colors are odd, this is a 

rainbow additive labeling of G n and so ra(G n ) = 2. We saw that ra(K 3 ) = 3. Actually, ra(K n ) = 3 for every integer n ≥ 3. Rainbow additive labelings showing that ra(K 4 ) ≤ 3 and ra(K 5 ) ≤ 3 are given in Figure 5.5. We will see in the next example that ra(K 4 ) = ra(K 5 ) = 3. Proposition 5.2 For every integer n ≥ 3, ra(K n ) = 3. Proof First, we show that ra(K n ) ≥ 3 for each integer n ≥ 3. Suppose that there exists a rainbow additive labeling of K n using only the labels 1 and 2 for some integer n ≥ 3. Let G be the spanning subgraph of K n whose edges are those labeled 1. Since no graph is irregular, there are two vertices u and v of G for which degG u = degG v = k for some integer k. However then, the vertices u and v are both colored 1 · k + 2(n − 1 − k) = 2n − k − 2 in K n , a contradiction. It therefore remains to show that there is a rainbow additive labeling of every complete graph K n , n ≥ 3, with the integers 1, 2, 3. We have already seen this with K 3 , K 4 , and K 5 . The labeling of K 4 is obtained from that of K 3 by adding a new vertex to K 3 and labeling each edge incident with this new vertex 1. The labeling of K 5 is obtained from that of K 4 by adding a new vertex to K 4 and labeling each edge incident with this new vertex 3. We continue this, alternating 1 and 3, as above. The vertex colors of K 5 are distinct integers in the range [6, 12]. Obtaining a labeling of K 6 in this manner, we have distinct vertex labels in the range [5, 13]. We can then proceed by induction to show that there is a rainbow additive labeling of K n for all n ≥ 5 such that the vertex colors are in the range [n + 1, 3n − 5] if n is odd and are in the range [n − 1, 3n − 3] if n is even. 

By the same observation made in the proof of Proposition 5.2, it can be seen that ra(G) ≥ 3 for every regular graph G of order 3 or more. There is also an infinite

5.1 Rainbow Additive Labelings

57

class of regular graphs G for which ra(G) = 4. In particular, ra(K r,r ) = 4 for every odd integer r ≥ 3. To illustrate this, we verify the following. Proposition 5.3 ra(K 3,3 ) = 4. Proof First, we show that there is no rainbow additive labeling of G = K 3,3 with the integers 1, 2, 3. Assume, to the contrary, that there is such a labeling f . If we define a function f 1 on E(G) by f 1 (e) = f (e) − 2 for each edge e of G, then f 1 is a rainbow additive labeling of G with the integers −1, 0, 1. Let U = {u 1 , u 2 , u 3 } 3 3    and W = {w1 , w2 , w3 } be the partite sets of G. Since f 1 (u i ) = f 1 (wi ), it follows that



i=1

i=1

f 1 (v) is even. Since the set of vertex colors is a 6-element subset

v∈V (G)

of S = [−3, 3] and S contains four odd integers, the only integer that is not a vertex color of G is an even integer. In particular, −3, −1, 1, 3 are all vertex colors and −3 and 3 are colors of vertices in the same partite set, say f 1 (u 1 ) = −3 and f 1 (u 2 ) = 3. Therefore, every edge incident with u 1 is labeled −1 and every edge incident with u 2 is labeled 1. If there are two edges of u 3 having the same label, then two vertices of W have the same color, which cannot occur. Thus, the three edges incident with u 3 are labeled −1, 0, 1 and so f  (u 3 ) = 0. This, however, implies that f 1 (w) = 0 for some vertex w ∈ W , which is a contradiction. Since there is no rainbow additive labeling of G with −1, 0, 1, there is also none with 1, 2, 3. As shown in Figure 5.6, there is a  rainbow additive labeling of G with 1, 2, 3, 4, however. Therefore, ra(K 3,3 ) = 4.

The rainbow additive indices of all regular complete bipartite graphs were determined in [6, 21]. Theorem 5.1 For each integer r ≥ 2,  ra(K r,r ) =

3 i f r is even 4 i f r is odd.

With the single exception of the graphs K r,r where r ≥ 3 is odd, all regular complete multipartite graphs have rainbow additive index 3 (see [13]). Theorem 5.2 For every regular complete k-partite graph G with k ≥ 3, ra(G) = 3. Fig. 5.6 A rainbow additive labeling of K 3,3

...... .. ...... ....... ......... ..... ....... ..... ....... . ... . ..... ... . ..... ..... ................... ..... ... .......... ....... ......... ....... ................. ..... ................. ......... . . . . . . . . . . . ..... ..... ....... ... .. .... . . . . . . ....... ......... . ..... . . . . . ............ ... ... ......... ..... ..... . ..... .. ... ......... ..... ..... ............. .... ............ ......... ..... ..... ........ ....... .......... ..... ......... ........ ........... ........ . . . . . .. .. .. . .. ..... .... ..... ..... ....... .... ............. ..... ......... ............ ....... ........ ......... ... .... ..... .......... . .. ... ..... ..... ......... ................. ...... ........ . ..... . ..... . . . . . . . . . . . . ....... ..... . . .... .. ... ....... ................ ..... ................ ........ ....... ......... ............. .......... ... ... ..... ..... ... ..... ..... ... ...... . ....... . .... .... . . ..... .... ..... ...... ......... ...... .......

4

2

K3,3 :

1

3

5

1

12

1

4

4

1

8

6

7

4

58

5 Additive Labelings

Fig. 5.7 A rainbow additive labeling of K 3,4

............ ............ ... .... ... ....... .. .... .. .... ........... .. ..... ..... ... ........ ... ..... ..................... .............. ....... ..................... ......... . . . . . . . . . . . ........ ....... ..... ... ....... ... ... ... . . . . . . . ..... . . . . . . . . . . ... ...... ........ ........ ..... .. ..... ........... ........ ... ..... ... ..... .. ..... ... .... ... ........................ ..... ..... .............. .... ..... . . . . . . . . ..... . . . . . . . . . . . . . ......... ....... ......... ... ..... . . . . . . . . . . . . . . . . . . . . . . ..... . ........... ... ..... ... ... ......... . . . . . . . . . . . . ..... . . . . . . ........ ...... ... .. . ... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ........ ...... .... .... ............... ...... ......... .............. ..... ............... . . . . . . . . . ............. .. . ... ... .... .... . . . . . . ... ..... ..... ..... ..... ..... ..... ..... . . . . . . . . . . . . .... ... ........ ........... ........ . ... ... ......... ........ ... ......... ... . . . . . . . . . ......... ... .... .. ......... ........ ... ... ......... ........ ... ... ......... ........ ... ... ......... ........ ......... ...................... .............. . .. ........... .... .......... ................

8

7

1

3

2

1

K3,4 :

3

1

4

2

1

1

1

2 5

3

9

3

6

While the rainbow additive index of K 3,3 is 4, the rainbow additive index of K 3,4 is smaller, namely, ra(K 3,4 ) = 3. An appropriate labeling of K 3,4 is shown in Figure 5.7. In fact, ra(K 3,s ) = 3 for s ∈ [4, 8]. While we have seen that ra(G) is either 3 or 4 for every regular complete multipartite graph G, there is no positive integer constant c such that ra(G) ≤ c for complete multipartite graphs G in general. In fact, for a fixed integer r ≥ 2, lim ra(K r,s ) = ∞.

s→∞

The value of the rainbow additive index of complete multipartite graphs that are not regular is an open problem. Problem 5.1 Determine the value of ra(G) for complete multipartite graphs G that are not regular. For all cycles and paths, the rainbow additive index has been determined. Theorem 5.3 ([13]) For an integer n ≥ 3, ⎧ n+1 ⎨ 2 if n ≡ 1 (mod 4) ra(Cn ) = n+2 if n is even 2 ⎩ n+3 if n ≡ 3 (mod 4). 2

(5.1)

Theorem 5.4 ([6]) For an integer n ≥ 3, ra(Pn ) =

⎧ ⎨ ⎩

n 2 n+1 2 n+2 2

if n ≡ 0 (mod 4) if n is odd if n ≡ 2 (mod 4).

It is immediate that the rainbow additive index of a star of order n ≥ 3 is n − 1. Not only do paths and stars of large order have large rainbow additive index, but this is true for all trees of large order.

5.1 Rainbow Additive Labelings

.......... .... ...... .... ..... ........ ....... ..... ..... ..... ................ . ... ..... . ..... .......... ...... ..... ..... ..... .................. ... .. . .. .. ....... . . . . . . . . . . ... .............. ... ... ..... ................ . ... . ..... .... ...... .......... ... ...... .................. ..... . .... ..... .........

6

1

7

2

.......... .... ...... .... ..... ....... ......

.......... .... ...... .... ..... ...... .......

7

7

8

7

...... ..... ....... . ..... ..... ...... ......

20

7

....... ..... ....... ..... . ..... ....... .....

2

1

7

21

14

................. .. ..... ..... ....... ......

.. ....... ......... ..... . .... ..... .........

6

13

59 3

3

...... ..... ....... . ..... ..... ...... .......

6

....... ..... ....... .... . ..... ..... .......

19

6

15

11

7

7

........ .... ....... .... . ..... ...... ......

4

4

10

9

.. ....... ......... ..... . .. . ................... . . . ... . . . . .. ..... ..... ..... ........ .. ... ......... ........ . . . . .. .... ..... ..... .............. .... ...... .... . ..... ............... ...... ........ ................ .. ...... ... . ..... ............ ...... ....... ............ ......... ... ... .... ..... ...... ......

2

16

7

12

5

7

Fig. 5.8 A tree T of order 19 with ra(T ) =

19+2 3

5

=7

Theorem 5.5 ([1, 6]) If T is a tree of order n ≥ 3 that is not a star, then n+2 ≤ ra(T ) ≤ n − 2. 3 The upper bound in Theorem 5.5 was established in [1], while the lower bound was determined in [6]. Both bounds are sharp. If T is a double star (a tree of diameter 3) of order n ≥ 4, then ra(T ) = n − 2. The tree T of Figure 5.8 shows that the lower bound is sharp when n = 19.

5.2 Proper Additive Labelings As we saw in the preceding section, a rainbow additive labeling of a graph G is an edge labeling of G that gives rise to a rainbow vertex coloring of G. There are also edge labelings of graphs that give rise to other types of vertex colorings. Indeed, rainbow vertex colorings are not even the most popular type of vertex coloring. Without a doubt, the best known vertex colorings are the proper colorings (where every two adjacent vertices are required to have different colors). Interest in these vertex colorings originated from attempts to solve the famous Four Color Problem. An additive labeling that results in a proper vertex coloring is a proper additive labeling. Since every connected graph of order 3 or more has a rainbow additive labeling, it also has a proper additive labeling. The minimum positive integer k for which a graph G has a proper additive labeling using elements from the set [k] is called the proper additive index of G, denoted by pa(G). Clearly, pa(G) ≤ ra(G) for every connected graph G of order 3 or more. The proper additive index of a graph G is therefore the minimum strength of a multigraph M obtained from G by possibly adding additional edges between adjacent vertices of G so that every two adjacent vertices of M have different degrees. If every two adjacent vertices of G already have different degrees, then pa(G) = 1. For example, pa(K r,s ) = 1 when r = s. See Figure 5.9 for the graph H1 = K 2,3 . If this is not the case, then pa(G) ≥ 2. The graph H2 = C4 , also shown in Figure 5.9,

60

5 Additive Labelings ................. .. .... ........ ........ ..... .............. .......... . . . . ..... ... . . ..... . . .. ..... ..... ..... ..... ...... ... ........... ..... ....... ...... ......... ..... ..... . . ..... . . . .. ... . . . ... ... ..... ...... ................ . . . . . ........ ..... ......... ..... ..... . . . ..... . .... ..... ..... ..... ..... ................. ......... ........ ...... ... . . ..... ..... .....

1

H1 :

2

3

1

1

2

1

2

1

3

1

.......... .... ...... .... ..... ...... .......

2

4

1 ... ...... ......... ..... . ..... ....... ......

2

3

1

1........................3........................... 2 ... ..

3

H2 : 2 ...... ..... ....... ..... .. ..... ...... .......

................. .. .. ...

..... ..... ....... .. ..... ..... ...... ......

H3 :

. ...... ............. .... ...... .... . . ..... ..... ........ ... ... ... ... ... ....... ..... ....... . ..... ..... ...... ......

...... ................ .. ..... ..... . ..... ....... ...... ... .. . ... ... .. ............... .... ... ... ................

4

2

2

1

4

5

3

Fig. 5.9 Graphs with proper additive indices 1, 2, and 3

has proper additive index 2. If one were to label the edges of H3 = C5 only with the labels 1 and 2, then there is at least one edge of H3 whose two neighboring edges have the same label, resulting in two adjacent vertices of the same color, which implies that pa(H3 ) ≥ 3. That pa(H3 ) = 3 is shown in Figure 5.9 as well. The interesting feature of proper additive labelings is that no example of a connected graph of order 3 or more has been found for which it is necessary to use the edge label 4. A popular conjecture (due to Michal Karo´nski, Tomasz Łuczak, and Andrew Thomason) resulting from this fact is the following, which goes by a rather catchy name. The 1-2-3 Conjecture If G is a connected graph of order 3 or more, then there is a proper additive labeling of G using the labels 1, 2, 3. Karo´nski, Łuczak, and Thomason [26] showed that this conjecture holds for all connected graphs of order 3 or more having chromatic number at most 3. Theorem 5.6 If G is a connected graph of order 3 or more having chromatic number 2 or 3, then pa(G) ≤ 3. While it has not been shown that pa(G) ≤ 3 for every connected graph G of order 3 or more, it has not even been shown that pa(G) ≤ 4 for such graphs. The following result, due to Maciej Kalkowski, Michal Karo´nski, and Florian Pfender [25], has been established, however. Theorem 5.7 If G is a connected graph of order 3 or more, then pa(G) ≤ 5.

5.3 Monochromatic Additive Labelings In addition to rainbow colorings and proper colorings, the third popular coloring in graph theory is a monochromatic coloring, which often occurs with edge colorings in Ramsey Theory. A monochromatic additive labeling of a graph G is an additive labeling that results in a vertex coloring in which all vertices of G have the same color. A monochromatic additive labeling of a graph G is therefore equivalent to adding additional edges between adjacent vertices of G, if necessary, to produce a

5.3 Monochromatic Additive Labelings

61

........ ................. .... ....... .... . ..... . .......... ..... ......... ...... ..... .................... .............. . . . . . ....... ..... .... . ....... ..... . . . .................. ..... ....... .......... ... ..... .. ..... .............. . .. . . . ..... ... .... .... . . . . . . . . . . . ..... .. ......... .. ..... ....... ..... ..... .......... ....... . ....... . . . . . . . . . . . . . . . . .... .... ....... ... .. .......... ..... ..... .. ...... ....... ..... ....... .... ......

6

3

6

2

1

6

2

6

6

3

...... ..... ....... .....

1

2

................. . ..... ... ..................... . . . . . . . . . . . . . ..... ... ..... . . . . . . . . . . . . . ..... ............. ..... ......... .......... ..... ....... . ..... . .......... ..... ....... ..... .......... . ..... ....... . . . ..... .. ....... ..... ..... ....... ..... ............ ..... ....... ...... ... ....... ................... .. ....... . .. .... ..... . . ..... ....... ..... ..... ...... ......

1................... .........6................

1

4

6

3

1

6

3

1

6

6

2

Fig. 5.10 A regular graph with a monochromatic additive labeling

regular multigraph. If G itself is regular, any labeling that assigns the same positive integer to every edge of G is a monochromatic additive labeling. A 1-factor of a graph G is a spanning 1-regular subgraph of G. A graph G is said to be 1-factorable if there are pairwise edge-disjoint 1-factors F1 , F2 , . . . , Fr of G such that E(G) = ∪ri=1 E(Fi ). Indeed, if G is a 1-factorable r -regular graph the label with 1-factors Fi (1 ≤ i ≤ r ), then the labeling that assigns each edge of Fi ai ∈ N is a monochromatic additive labeling in which every vertex color is ri=1 ai . A regular graph that is not 1-factorable may very well have a monochromatic additive labeling in which not all edges are labeled the same (see Figure 5.10). A major difference between monochromatic additive labelings and rainbow or proper additive labelings is that not all non-regular graphs have a monochromatic additive labeling. For example, no tree of order 3 or more has such a labeling and K r,s with r = s also has no such labeling. If G is a graph possessing a monochromatic additive labeling, then the minimum positive integer k for which G has a monochromatic additive labeling using elements from the set [k] is called the monochromatic additive index of G, denoted by ma(G). Therefore, ma(G) = 1 if and only if G is regular. While no non-regular complete bipartite graph has a monochromatic additive labeling, a non-regular complete k-partite graph with k ≥ 3 may have a monochromatic additive labeling. For example, the graph K 2,3,4 has a monochromatic additive labeling. To see this, let V1 , V2 , V3 be the partite sets of K 2,3,4 where |V1 | = 2, |V2 | = 3, and |V3 | = 4. For i, j ∈ {1, 2, 3} and i = j, let [Vi , V j ] denote the set of edges joining a vertex of Vi and a vertex of V j . Define an edge labeling f of K 2,3,4 by assigning the label 4 to each edge in [V1 , V2 ], the label 9 to each edge in [V1 , V3 ], and the label 10 to each edge in [V2 , V3 ] (see Figure 5.11). Since the color of every vertex of K 2,3,4 is 48, it follows that f is a monochromatic additive labeling of K 2,3,4 . As another example, the graph K 3,4,5 also has a monochromatic additive labeling. Let V1 , V2 , V3 be the partite sets of K 3,4,5 with |V1 | = 3, |V2 | = 4, and |V3 | = 5. Define an edge labeling f of K 3,4,5 by assigning the label 5 to each edge in [V1 , V2 ], the label 8 to each edge in [V1 , V3 ], and the label 9 to each edge in [V2 , V3 ] (see Figure 5.11). Since the color of every vertex of K 3,4,5 is 60, it follows that f is a monochromatic additive labeling of K 3,4,5 . Whether there is a monochromatic additive labeling of these two graphs where the labels of the edges joining vertices in each pair of partite sets are not a constant is not known. In any case, the monochromatic additive labelings in Figure 5.11 show that ma(K 2,3,4 ) ≤ 10 and ma(K 3,4,5 ) ≤ 9.

62

5 Additive Labelings ........ . . ... .. .... 1 ..... ........ .... ........ ... ..... .. . . . . . . . . . . . . ... ... . ....... ..... ......... . ..... . .... .....

V

K2,3,4 :

4

qqq qqq 9 qqq qq

.... ..... . ... ..... ......... .. . ....... ..... .... ..... ........... ..... ........ ..... ..... .. .. .. . ....... .......... . ... . ..... .. ... ...

..... .... . . .... . .... ...... ............. ..... 2 .. ..... . ...... ..... ..... ......... ..... ..... ....... ..... ..... ......... .... ..... ... . .. ...

V1

V

10

q qq qq q q

........ ..... ..... ..... ..... ..... ..... ..... ..... ..... . ...... ............. .. .............. .............. ... .......... ... . . . ..... ..... ..... ..... ..... ..... ..... ..... ..... .......

K3,4,5 :

V3

5

qqq qqq 8 qqq qqq

...... ..... ... ........ ........ .. .. 2 .. ........ ... ..... . ... .... ..... ............. .... . ..... ....... .... . . ..... ........ ..... ..... ......... . . . ..... ..... ..... . .....

V

q qqq qq qq qq

9

........ ..... ..... ..... ..... ..... ..... ..... ..... ......... .. ....... ......... .......... .............. .............. .... ....... ...... ... ........ .. ...... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..

V3

Fig. 5.11 Monochromatic additive labelings of K 2,3,4 and K 3,4,5

The graphs K 2,3,4 and K 3,4,5 belong to a class of complete 3-partite graphs described in the following example. Proposition 5.4 The complete 3-partite graph K r,s,t with 2 ≤ r ≤ s ≤ t has a monochromatic additive labeling if and only if t < r + s. Proof Let V1 , V2 , V3 be the partite sets of G = K r,s,t where |V1 | = r , |V2 | = s, and |V3 | = t. Suppose that G has a monochromatic additive labeling f . Then f produces a monochromatic vertex coloring of G in which each vertex color is some positive integer p. The set of edges incident with the vertices of V3 is [V1 ∪ V2 , V3 ] and the set of edges incident with the vertices of V1 ∪ V2 is [V1 , V2 ] ∪ [V1 ∪ V2 , V3 ]. Since the sum of the labels of the edges incident with the vertices of V3 is pt and the sum of the labels of the edges in [V1 ∪ V2 , V3 ] incident with the vertices of V1 ∪ V2 is less than r p + sp, it follows that t p < (r + s) p and so t < r + s. It remains to show that if t < r + s, then G has a monochromatic additive labeling. Since t < r + s, it follows that r + s − t is a positive integer. Define an edge labeling f of G by assigning the label a = t (r + s − t) to each edge in [V1 , V2 ], the label b = s(r + t − s) to each edge in [V1 , V3 ], and the label c = r (s + t − r ) to each edge in [V2 , V3 ]. Since the resulting color of every vertex of G is 2r st, it follows that f is a monochromatic additive labeling of G. If d = gcd(a, b, c), then multiplying each edge label by 1/d results in another monochromatic additive labeling of G. This implies that ma(G) ≤ r (s − r + t)/d. 

The complete 3-partite graphs considered in Proposition 5.4 have another wellknown property. A path in a graph G is a Hamiltonian path if it contains every vertex of G. A graph G is Hamiltonian-connected if every two vertices of G are connected by a Hamiltonian path. A Hamiltonian cycle in a graph G is a cycle containing every vertex of G. A graph having a Hamiltonian cycle is a Hamiltonian graph. For a complete k-partite graph G = K n 1 ,n 2 ,...,n k , where k ≥ 3 and n 1 ≤ n 2 ≤ k−1  · · · ≤ n k , it is well known that G is Hamiltonian if and only if n k ≤ n i and G i=1

is Hamiltonian-connected if and only if n k
Δ(G)(n − 1)/2. The following result is well known (see [9, p. 258], for example). Theorem 6.12 If G is an overfull graph, then G does not possess a proper Δ(G)coloring of its edges.

6.3 Zonal Labelings of Cubic Maps

81

The following argument is similar to the one used in the proof of Theorem 6.10. Theorem 6.13 If G is a connected cubic plane graph with bridges, then G is not zonal. Proof Assume, to the contrary, that there is a connected cubic plane graph G with bridges possessing a zonal labeling . Let e = uv be a bridge of G such that v belongs to an end-block B of G. Necessarily, the order of B is at least 5 and v is incident with two edges of B, say vx and vy. According to the label (v) of v, we may assign the colors 1, 2, 3 to the three edges incident with v such that the type of v is (v). Next, we show that the color c(e) of e and the zonal labeling  of G uniquely determines a proper 3-coloring c of the edges of B such that the type of each vertex v of B is (v ). Let f be an arbitrary edge of B where f is incident with a vertex w of B. Since the color of each edge incident with v is uniquely determined by c(e) and (v), we may assume that f is not incident with v. Since B is 2-connected, there is a cycle C of B containing both v and w. Then xv and vy are two adjacent edges of C. First, suppose that C is the boundary cycle of some zone of B. By Observation 6.9, the zonal labeling  restricted to the vertices of C is a zonal labeling of C. Since the color of the edge vx of C is given, it then follows by Corollary 6.1 that the colors of all edges of C are uniquely determined. Thus, the colors of the two edges of C incident with w are uniquely determined, as is the color of the edge f . Hence, we may assume that C is not the boundary cycle of any zone of B. Therefore, C encloses more than one zone of B. Let C (1) be the boundary cycle of a zone R (1) of B lying within C such that v lies on C (1) . The zonal labeling  restricted to the vertices of C (1) is a zonal labeling of C (1) . By Observation 6.9 and Corollary 6.1, the colors of the edges of C (1) are uniquely determined. There exists a sequence C (1) , C (2) , . . . , C (k) of cycles, k ≥ 2, where C (i) is the boundary cycle of a zone R (i) lying within C such that C (i) and C (i+1) have an edge in common for 1 ≤ i ≤ k − 1 and w is a vertex on C (k) . Since the colors of the edges of C (1) are uniquely determined and an edge of C (1) belongs to C (2) , the colors of the edges of C (2) are uniquely determined. Continuing in this manner, we see that the colors of the edges of C (k) are uniquely determined as well, as is the color of the edge f . Thus, there is a proper 3-coloring of the edges of B. Let p ≥ 5 be the order of B. Since B has p − 1 vertices of degree 3 and one vertex of degree 2, it follows that p is odd and the size of B is 3 p2− 1 > 3( p2− 1) = Δ(B)(2p − 1) and so B is a overfull graph. By Theorem 6.12, there is no proper 3-coloring of the edges of B. This is a contradiction.  The following corollary is a consequence of Corollary 6.2 and Theorem 6.13, which establishes the truth of Conjecture 6.1. Theorem 6.14 A connected cubic plane graph G is zonal if and only if G is bridgeless.

82

6 Zonal Labelings

6.4 Inner Zonal Labelings of Bicubic Maps The argument given that establishes the truth of Theorem 6.11 makes use of the fact that we knew that the Four Color Theorem is true. It would be considerably more interesting if it could be shown that every cubic map is zonal without using the Four Color Theorem (and without using computers). We state this fundamental question next. Problem 6.2 Can every cubic map be proved to be zonal without using the Four Color Theorem or computers? Of course, we were able to establish the zonality of cycles without using the Four Color Theorem. Every cycle is a subgraph and the boundary of a zone of some cubic map. This brings up the question of the existence of certain types of labelings of graphs that are subgraphs of some cubic map. By a bicubic map B is meant a 2-connected planar graph embedded in the plane all of whose vertices have degree 2 or 3, where the boundary of the exterior region of B is a cycle C and every vertex lying interior to C has degree 3. Therefore, if C is a cycle in a cubic map M and H is the submap of M consisting of C and all vertices and edges of M lying interior to C, then H is a bicubic map. If no vertex lies interior to C, then either H = C or H is C with chords; while if every vertex of H has degree 3, then H = M. This suggests a labeling closely related to a zonal labeling. Let B be a bicubic map. A labeling of the vertices of B with the elements 1 and 2 of Z3 is an inner zonal labeling if the label of each interior zone of B is 0. If B contains an inner zonal labeling, then B is inner zonal. Therefore, the labeling of K 4 − e given in Figure 6.4 is an inner zonal labeling and so K 4 − e, while not zonal, is inner zonal. Of course, if a bicubic map B has an inner zonal labeling such that the label of the exterior zone is 0 as well, then the labeling is zonal and B itself is zonal. Therefore, if B is a cycle or a cubic map, then B is inner zonal. The question is which bicubic maps are inner zonal and which of these can be shown to have this property—without using the Four Color Theorem or computers. There is an observation for inner zonal labelings of bicubic maps that is the analogue of Observation 6.1. Observation 6.15 If  is an inner zonal labeling of a bicubic map, then so too is its complementary labeling . Lemma 6.1 Let C be an n-cycle, n ≥ 4, and let u 1 v1 and u 2 v2 be two nonadjacent edges of C. There exists a zonal labeling  of C such that (u i ) = (vi ) for i = 1, 2. Proof If n = 2k ≥ 4, then we may label k vertices of C (including u 1 and v1 ) the color 1 and the remaining k vertices of C (including u 2 and v2 ) the color 2. If n = 2k + 1 ≥ 5, then we may label k + 2 vertices of C (including u 1 , v1 , u 2 , and v2 ) the color 1 and the remaining k − 1 vertices of C the color 2. In each case, we have a zonal labeling of C. 

6.4 Inner Zonal Labelings of Bicubic Maps

83

Actually, every bicubic map is inner zonal. To see this, let G be a bicubic map and let G  be another copy of G embedded in the plane in the same way that G is. For each vertex v of G, let v be the vertex of G  corresponding to v. Then a plane graph H can be constructed from G and G  by adding the edge uu  for each vertex u of G having degree 2. Thus, H is a cubic map and by Theorem 6.8, H has a zonal labeling . This labeling , restricted to G, is an inner zonal labeling of G. However, the reason we know that H has a zonal labeling is because there is a proper 3-coloring of the edges of H and the reason we know this is because the Four Color Theorem is true. And, the reason we know that the Four Color Theorem is true is because there is a computer-aided proof of this theorem. This brings up the question as to whether there exists an independent proof of the inner zonality of every bicubic map, namely, one that does not require prior knowledge of the Four Color Theorem. We now give such a proof for one class of bicubic maps. Theorem 6.16 Let G be a bicubic map where the boundary of the exterior region is a Hamiltonian cycle C with k pairwise nonadjacent chords for some positive integer k. There exists an inner zonal labeling  of G where (x) = (y) for each chord x y of C. Proof We proceed by induction on k. First, suppose that G is a bicubic map where the boundary of the exterior region is a Hamiltonian cycle of G with one chord x y. Let R1 and R2 be the two interior zones of G. Label x and y with the color 1. If the boundary C1 of R1 is a triangle, then we label the remaining vertex on C1 the color 1. If C1 is an m-cycle for some integer m ≥ 4, then, by Lemma 6.1 and Observation 6.15, there exists a zonal labeling 1 of C1 such that 1 (x) = 1 (y) = 1. Similarly, there is a zonal labeling 2 of the boundary of R2 such that 2 (x) = 2 (y) = 1. We now define an inner zonal labeling  of G by (w) = 1 (w) if w ∈ V (C1 ) and (w) = 2 (w) if w ∈ V (G) − V (C1 ). Then (x) = (y). Thus, the basis step of the induction holds. Assume that the statement is true for all bicubic maps in which the boundary of the exterior region is a Hamiltonian cycle with k pairwise nonadjacent chords for some positive integer k. Let G be a bicubic map in which the boundary of the exterior region is a Hamiltonian cycle C with k + 1 pairwise nonadjacent chords. Let R1 be an interior zone of G such that the boundary C1 of R1 contains only one chord uv of C. Deleting V (C1 ) − {u, v} from G produces a bicubic map G  in which the boundary of the exterior region is a Hamiltonian cycle C  with k pairwise nonadjacent chords. By the induction hypothesis, there exists an inner zonal labeling  of G  such that  (s) =  (t) for each chord st of C  . Next, we show that the bicubic map G has an inner zonal labeling  such that ( p) = (q) for every chord pq of C. We consider two cases, according to whether  (u) =  (v) or  (u) =  (v). Case 1.  (u) =  (v). By Observation 6.15, we may assume that  (u) =  (v) = 1. If C1 is a triangle, then we label the remaining vertex of C1 the color 1. Thus, we may assume that C1 is an m-cycle, where m ≥ 4. By Lemma 6.1 and Observation 6.15, there exists a zonal labeling 1 of C1 such that 1 (u) = 1 (v) = 1. Then an inner zonal labeling  of G can be constructed from the labelings  and 1 by defining (w) = 1 (w) if w ∈ V (C1 ) and (w) =  (w) otherwise.

84

6 Zonal Labelings

Case 2.  (u) =  (v). Let R2 be the interior zone of G that is adjacent to R1 . Thus, R2 contains uv on its boundary as well as another chord x y of G. It then follows by the defining property of  that  (x) =  (y). Let C2 be the boundary cycle of R2 . Then uv and x y are two nonadjacent edges of C2 . By Lemma 6.1 and Observation 6.15, there exists a zonal labeling 2 of C2 such that 2 (u) = 2 (v) and 2 (x) = 2 (y) =  (x) =  (y). We now define another inner zonal labeling ∗ of G  by ∗ (w) = 2 (w) if w ∈ V (C2 ) and ∗ (w) =  (w) if w ∈ V (G  ) − V (C2 ). Then ∗ is an inner zonal labeling of G  where ∗ (s) = ∗ (t) for each chord st of C  and ∗ (u) = ∗ (v). We now proceed as in Case 1 to construct an inner zonal labeling  of G. In each case, the inner zonal labeling  of G has the property that  (s) =  (t) for each chord st of C. Therefore, the theorem follows by the Principle of Mathematical Induction.  By Theorem 6.16, every bicubic map G in which the boundary of the exterior region is a Hamiltonian cycle C is inner zonal. If G = C, then G is zonal. Thus, we have the following corollary, the truth of which is independent of the Four Color Theorem. Corollary 6.3 Every plane graph G with Δ(G) ≤ 3 where the boundary cycle of the exterior zone is a Hamiltonian cycle of G is inner zonal.

References

1. M. Aigner, E. Triesch, Irregular assignments of trees and forests. SIAM J. Discret. Math. 3, 439–449 (1990) 2. L.W. Beineke, Characterizations of derived graphs. J. Comb. Theory 9, 129–135 (1970) 3. C. Berge, Théorie des Graphes et Ses Applications (Dunod, Paris, 1958) 4. N.L. Biggs, E.K. Lloyd, R.J. Wilson, Graph Theory 1736–1936 (Clarendon Press, Oxford, 1976) 5. A. Cayley, A theorem on trees. Q. J. Math. 23, 376–378 (1889) 6. G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, F. Saba, Irregular networks. Congr. Numer. 64, 197–210 (1988) 7. G. Chartrand, F. Okamoto, P. Zhang, The sigma chromatic number of a graph. Graphs Comb. 26, 755–773 (2010) 8. G. Chartrand, L. Lesniak, P. Zhang, Graphs & Digraphs, 6th edn. (Chapman & Hall/CRC, Boca Raton, FL, 2015) 9. G. Chartrand, P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC Press, Boca Raton, 2009) 10. T. Deretsky, S.M. Lee, J. Mitchem, On vertex prime labelings of graphs, in Graph Theory, Combinatorics, and Applications (Wiley-Interscience, New York, 1991), pp. 359–369 11. P. Erd˝os, R.J. Wilson, On the chromatic index of almost all graphs. J. Comb. Theory Ser. B 23, 255–257 (1977) 12. W. Fang, A computational approach to the graceful tree conjecture (2010). arXiv: 1003.3045v2 13. R.J. Faudree, M.S. Jacobson, J. Lehel, R.H. Schelp, Irregular networks, regular graphs and integer matrices with distinct row and column sums. Discret. Math. 76, 223–240 (1989) 14. M. Fiedler (ed.), Theory of Graphs and its Applications, in Proceedings of the Symposium Held in Smolenice in June 1963. Prague, Pub. House of the Czechoslovak Academy of Sciences (New York, Academic Press, 1964) 15. R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Math. 6, 239–250 (1938) 16. R. Frucht, Graceful numbering of wheels and related graphs. Ann. N. Y. Acad. Sci. 319, 219– 229 (1979) 17. H.L. Fu, K.C. Huang, On prime labellings. Graph Theory Appl. Discret. Math. 127, 181–186 (1994) 18. J.A. Gallian, A dynamic survey of graph labeling. Electron. J. Comb. (2017) (#DS6) 19. S.W. Golomb, How to number a graph, Graph Theory and Computing (Academic Press, New York, 1972), pp. 23–37 20. R.L. Graham, N.J.A. Sloane, On additive bases and harmonious graphs. SIAM J. Alg. Disc. Math. 1, 382–404 (1980) 21. A. Gyárfás, The irregularity strength of K m,m is 4 for odd m. Discret. Math. 71, 273–274 (1988) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6

85

86

References

22. P. Haxell, O. Pikhurko, A. Taraz, Primality of trees. J. Comb. 2, 481–500 (2011) 23. P. Hrnˇciar, A. Haviar, All trees of diameter five are graceful. Discret. Math. 233, 133–150 (2001) 24. C. Huang, A. Kotzig, A. Rosa, Further results on tree labellings. Util. Math. 21, 31–48 (1982) 25. M. Kalkowski, M. Karo´nski, F. Pfender, Vertex-coloring edge-weightings: towards the 1-2-3conjecture. J. Comb. Theory Ser. B 100, 347–349 (2010) 26. M. Karo´nski, T. Łuczak, A. Thomason, Edge weights and vertex colours. J. Comb. Theory Ser. B 91, 151–157 (2004) 27. M. Kneser, Aufgabe 300. Jahresber. Deutsch., Math. Verein. 58, 27 (1955) 28. D. König, Theorie der endlichen und unendliehen Graphen (Akademische Verlagsgesellschaft, Leipzig, 1936) 29. A. Kotzig, A. Rosa, Magic valuations of finite graphs. Canad. Math. Bull. 13, 451–461 (1970) 30. S.M. Lee, I. Wui, J. Yeh, On the amalgamation of prime graphs. Bull. Malays. Math. Soc. 11, 59–67 (1988) 31. L. Lovász, Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory Ser. A 25, 319–324 (1978) 32. T.A. McKee, F.R. McMorris, Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications (Philadelphia, PA, 1999) 33. J.W. Moon, Counting labeled trees, in From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress (Vancouver, 1969) (1970). (Canadian Mathematical Monographs, Canad. Math. Congress, Montreal, Que) 34. O. Ore, Theory of Graphs (Providence, RI, 1962). (Am. Math. Soc. Colloq. Pub) 35. J. Petersen, Die Theorie der regulären Graphen. Acta Math. 15, 193–220 (1891) 36. H. Prüfer, Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. 27, 142–144 (1918) 37. S.N. Rao, Prime labeling, in R. C. Bose Centenary Symposium on Discrete Mathematics. and Applications, Kolkata (2002) ˇ 38. G. Ringel, Problem 25, in Theory of Graphs and its Applications (Nakl. CSAV, Prague, 1964), p. 162 39. G. Ringel, J.W.T. Youngs, Solution of the Heawood map-coloring problem. Proc. Nat. Acad. Sci. USA 60, 438–445 (1968) 40. A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Gordon and Breach, New York, 1967), pp. 349–355 41. J. Sedláˇcek, Problem #27, Theory of Graphs and its Applicationsin, Proceedings of the Symposium Held in Smolenice in June 1963. Prague, Pub. House of the Czechoslovak Academy of Sciences; (New York, Academic Press, 1964) 42. J. Sedláˇcek, On magic graphs. Math. Slov. 26, 329–335 (1976) 43. M.A. Seoud, A.T. Diab, E.A. Elsakhawi, On strongly-C harmonious, relatively prime, odd graceful and cordial graphs. Proc. Math. Phys. Soc. Egypt 73, 33–55 (1998) 44. S. Stahl, n-Tuple colorings and associated graphs. J. Comb. Theory Ser. B 20, 185–203 (1976) 45. M.A. Seoud, M.Z. Youssef, On prime labelings of graphs. Congr. Numer. 141, 203–215 (1999) 46. N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences (OEIS). Sequence Number A213273 47. B.M. Stewart, Magic graphs. Canad. J. Math. 18, 1031–1059 (1966) 48. B.M. Stewart, Supermagic complete graphs. Canad. J. Math. 19, 427–438 (1967) 49. P.G. Tait, Remarks on the colouring of maps. Proc. Royal Soc. Edinburgh 10(729), 501–503 (1880) 50. R. Tout, A.N. Dabboucy, K. Howalla, Prime labeling of graphs. Nat. Acad. Sci. Lett. 5, 365–368 (1982) 51. V.G. Vizing, On an estimate of the chromatic class of a p-graph. (Russian) Diskret. Analiz. 3, 25–30 (1964) 52. R. Wilson, Four Colors Suffice: How the Map Problem Was Solved (Princeton University Press, Princeton, NJ, 2002) 53. P. Zhang, Color-Induced Graph Colorings (Springer, New York, 2015) 54. P. Zhang, A Kaleidoscopic View of Graph Colorings (Springer, New York, 2016)

Index

Symbols 1-2-3 Conjecture, 60 1-factor, 61 1-factorable, 61 2-prime integer, 45 2-prime labeling, 45 2-subset, 47, 48 2-subset labeling, 47, 48 α-valuations, 8 β-valuations, 8 γ -valuations, 8 ρ-valuations, 8 k-prime integer, 48 k-prime labeling, 48 k-subset, 49 k-subset graph, 51 k-subset labeling, 49 k-tuple coloring, 51 kth power, 11

A Additive basis, 21 Additive labeling, 54 Antimagic labeling, 55 Appel, Kenneth, 74

B Beineke’s Theorem, 46 Beineke, Lowell, 46 Berge, Claude, 7 Bertrand’s Postulate, 30 Bertrand, Joseph, 30 Bicubic map, 82 Biggs, Norman, 7 Bow tie graph, 14

C Cayley’s Tree Formula, 2 Cayley, Arthur, 1 Central vertices, 16 Chartrand, Gary, 55, 63 Chebyshev, Pafnuty, 30 Chromatic index, 4 Chromatic number, 3 Coloring, 3 Complementary coloring, 75 Complementary labeling, 10, 26, 72, 75 Composition graph, 36 Coprime, 29 Coprime index, 33 Coprime labeling, 33 Cube, 11 Cubic maps, 73 Cyclic decomposition, 12

D Diab, A. T., 32 Double star, 16

E Edge-labeled graph, 3 Egan, Cooroo, 72 Elsakhawi, E. A., 32 Entringer, Roger, 29, 32 Erd˝os, Paul, 16, 30 Euler, Leonhard, 7

F Four Color Problem, 74 Four Color Theorem, 63, 74

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 G. Chartrand et al., How to Label a Graph, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-16863-6

87

88

Index

Frucht, Roberto, 16

Luczak, Tomasz, 60

G Gallian, Joseph, 19 Gcd-rainbow index, 34 Gcd-rainbow labeling, 34 Golomb, Solomon, 8 Graceful graph, 9 Graceful labeling, 8, 9 Graceful Tree Conjecture, 18 Graham, Ronald, 22, 25 Graph labeling, 3 Guthrie, Francis, 73

M McKee, Terry A., 48 McMorris, F. R., 48 Monochromatic additive index, 61 Monochromatic additive labeling, 60 Monochromatic coloring, 4, 60 Monochromatic sigma chromatic number, 66 Monochromatic sigma labeling, 66 Monochromatic subgraph, 4 Moon, John, 2 Multi-prime index, 36 Multi-prime labeling, 35

H Haken, Wolfgang, 74 Hamiltonian cycle, 62 Hamiltonian graph, 62 Hamiltonian path, 62 Hamiltonian-connected, 62 Harmonious graph, 22 Harmonious labeling, 22 Harmonious Tree Conjecture, 25 Haxell, Penny, 32, 33 Heawood Map Coloring Problem, 18

I Inner zonal labeling, 82 Inner zonal map, 82 Intersection graph, 48 Irregular graph, 53 Irregularity strength, 55

K Kalkowski, Maciej, 60 Karo´nski, Michal, 60 Kneser graph, 39 Kneser’s Conjecture, 39 Kneser, Martin, 39 Konig, Dénes, 7 Kotzig’s Conjecture, 8 Kotzig, Anton, 8, 18

L Label of a zone, 70 Line graph, 46 Lloyd, E. Keith, 7 Lovász, László, 39 Lucky labeling, 64

N Nearly irregular graph, 53

O Odd graph, 39 Ore, Oystein, 7 Overfull graph, 80

P Petersen graph, 22, 38 Pfender, Florian, 60 Pikhurko, Oleg, 32, 33 Prüfer, Heinz, 2 Prime graph, 29 Prime labeling, 29 Prime Tree Conjecture, 32 Proper additive index, 59 Proper additive labeling, 59 Proper coloring, 3 Proper sigma labeling, 63

R Rainbow additive index, 55 Rainbow additive labeling, 54 Rainbow coloring, 4, 34, 54 Rainbow sigma chromatic number, 65 Rainbow sigma labeling, 65 Ramanujan, Srinivasa, 30 Rao, S. N., 33 Ringel’s Conjecture, 8 Ringel, Gerhard, 8, 18 Rosa, Alexander, 8, 14, 18

Index S Seoud, M. A., 32, 33 Sigma chromatic number, 63 Sigma labeling, 63 Sloane, Neil, 22, 25 Square, 11 Stahl, Saul, 51 Strength, 55 Subset labeling, 38

89 U Unicyclic graph, 33

V Vertex label, 4 Vertex labeling, 4 Vertex-labeled graph, 3

W Wilson, Robin, 7, 16 T Tait’s Theorem, 74 Tait, Peter, 74 Taraz, Anusch, 32, 33 Thomason, Andrew, 60 Tree, 1 Twins, 65 Type 1 vertex, 75, 78 Type 2 vertex, 75, 78

Y Youngs, J. W. T., 18 Youssef, M. Z., 33

Z Zonal graph, 72 Zonal labeling, 72